I'll try to be extremely fair so as not to mischaracterize any position, undoubtedly I will be corrected.
possibly partly because people often repeat that the differences between all the MSS aren't quite so big, so what's the fuss?
That has largely been my attitude as well. I've used the ESV in a NKJV congregation for a long while and have never even noticed the difference. My wife used a KJV in a different congregation (she doesn't remember their translation) and said she didn't notice a difference. However, I think it comes down to a question of whether it is the very word of God, or only 99% the word of God. God's word is precious and should be treated as such. I think both sides have a high view of Scripture but differ as to where that Scripture is contained and how God preserved it.
The Hebrew was transcribed incredibly meticulously, so there are relatively few readings, so textual criticism usually is about the Greek. The Greek was transcribed by many different people, presumably some for private use, some for public use and there are many more variations, but even unbelieving textual scholars are astounded by how few they are and how little they affect the text. The most critical has to concede that what we have today is extremely close to any ancient copy ever discovered, contrary to the claim of many unbelievers that it has become so corrupt as to be unrecognizable.
As a side note, it seems to me that there are several camps one could choose:
A perspective that God preserved his word in
all the extant copies.
A perspective that God preserved his word in the Greek churches especially, perhaps called the Byzantine Priority and derived from this mainly, the Majority text (a hypothetical text created by taking the readings that have the most manuscripts, regardless of antiquity).
A perspective that God preserved his word within certain Byzantine texts, used for the Reformation.
Note that all groups must perform some kind of textual criticism simply because no two manuscripts are exactly alike. Most of the variations are extremely minor and non-translatable. Many are slips of the pen, obviously a typo.
I think the problem with Warfield and his ilk, from what I'm starting to understand, was that the "Received Text" was, through God's providential preserving of the text of the NT, publically well known and accepted in the Church, but Warfield et al, were willing to give a place for a different family or families of MSS, with significant differences, some of MSS which had, for instance, been recently recovered from a waste-paper basket on St Catherine's Monastery, Sinai.
It's a little more nuanced than that but I think that is mostly a fair assessment from the "Received Text" (TR) side. TR advocates would say that God doesn't leave his church without his entire word, therefore, what the church had during the time of the reformation must have been what was providentially preserved. It is a little more nuanced because for years, though the TR was certainly the standard text, nearly every commentator or student would talk about "older manuscripts" or "the oldest and best". So though they believed what they had accurately represented the word of God, they also apparently believed there could still be mistakes in it, though it is certainly arguable they wouldn't have assumed that there were as many variants as Warfield did.
Though at this point it's a bit difficult to compare, because textual criticism was really in its infancy. Many truly seemed to have no idea how many variants they were until things like Walton's Polyglot (which collated all the known variants) appeared. Bahnsen said it is like judging 17th century doctor's knowledge of medicine using 21st century standards.
As a side note about the "trash can" story, note that Tischendorf's account says that he found some monks cleaning out some old manuscripts, among them some leaves from the Septuagint (Greek translation of the Old Testament). Tischendorf became very excited and asked for more and the monks clammed up. Several years later he came back and after presenting one of the monks with some of his work, the monk showed him, in a closet, wrapped in red cloth, the Codex Sinaiticus (some say that's an indication of honor). It could be said that the monks realized it's value and put it back to be stored safely, but it's not clear from Tischendorf's account that this was actually the case. This was in the early 1800s so not sure how "recent" you'd consider that.
By doing this Warfield wasn't treating the Received Text in a biblical way, as his approach and that of modern textual criticism, theoretically means that the settled text of the Bible which God has supposed to have preserved among us - as we learn from Scripture's view of itself, which view we must start and end with in textual studies - can be opened up at any time by new discoveries, thus undermining our confidence in what we have.
That seems to be the TR position, though I wonder if earlier generations, using a particular form of Scripture would have felt the same: that God had preserved his word in their current Greek texts and these TR folks had no right to mess with it using other data. Also note that (at least from my perspective) it is a little more nuanced than that because not all families of Greek texts were the same. Christians in different parts of the world had different Greek texts, which line was the preserved line? TR advocates would probably say the one in the Greek speaking church, where they were used the most and copied the most. That is probably reasonable if you assume one line of preservation. It's also a little bit tricky because the main place the Greek manuscripts were copied would have been in Greek-speaking churches, most others copied the Latin as far as I know, so naturally there are more copies in the Byzantine family. One could reasonably assume (from a TR preservationist standpoint) that they would also be the most accurate.
Theoretically, according to modernist textual criticism, there might be a Bible out there in the sands, very different to our own, which may yet be much more representative of the autographs. But that wouldn't be God preseving the text among us, bit allowing important textual material to languish in a place where few or none of God's people had access to it - which is incompatible with its preservation and perseverance among us.
I don't know if that's the case. Note that I'm not really a Critical Text (CT) advocate but I don't know that anyone would accept a different reading based on one manuscript. I do have some concerns that some readings are alleged to have been based off of one or two manuscripts in the CT, I have not studied these myself though. But comparing manuscripts can be a very tricky subject. Age alone isn't enough because you might have an old manuscript that has been copied a dozen times by incompetent transcribers, while another relatively recent copy (say, 11th century) was copied directly from an older copy that was copied directly from the autograph. So sometimes textual critics seem to think in terms of "generations", a "third generation" manuscript, even if copied much later, is of more importance than a twelfth generation manuscript, even if much older. So age, family, level of care, all can play a role in assigning a "reliability" to the copy. But note once again that Erasmus had to do something similar when he was working on the original edition of the TR.
But yes, I think that is a valid concern: that God's people would not have had the real words for most of their history.
Things become a little more tricky because there is not just one edition of the TR. Erasmus produced several (the one Luther used differed slightly from later editions), Stephanus produced slightly improved copies of Erasmus, and Beza likewise worked from them. Between these three there are some 200+ variations, aside from marginal notes that mentioned alternate readings. The KJV Bible seems to have used all three, plus other sources, and produced their translation. Scrivener, in 1894 produced a new Greek text that essentially compiled the supposed Greek underlying the KJV (sort of a back-translation). It is this TR that many accept today. Did God providentially preserve his word in this or the others? Perhaps, though the Reformers and Puritans didn't have any problem "correcting" at least some readings from other Greek manuscripts. I also am concerned that certain readings have little Greek support, especially Revelation 22:19 where the term "book" of life can't be found in any but two Greek manuscripts, and both of those possibly made as a copy of Erasmus' edition. Most TR advocates don't seem to see this as a problem however, because they believe it to be preserved: that is primary and evidence is secondary.
I think it was partially this that made Edward Hills' view attractive: Yes, the TR has variants, we can't be
absolutely certain which are the correct but it is this edition that gives us
maximum certainty. It has the fewest variants and was counted reliable for a huge portion of the Christian Church.
So to kinda sum up. The TR position (if one can call it a solitary position, since everyone seems to have slightly different reasons), is that God promised to preserve his word. We have his word, we believe his promises, therefore what we have is it. This certainly I could see as the "safe" route.
One of the problems I see with that position is that it seems that any Christian community before could have claimed the same thing before the TR, the Waldenses for example, other Greek-speaking communities, for whom the TR represented closely, but not perfectly what they had.
On the other hand, the CT position places a lot of weight on more ancient texts more recently discovered, and it is quite easy to introduce human bias. Part of me is excited at the effort to "purify" the text as much and get as close to the originals as possible, the other part is uneasy in that human knowledge is fallible. It is certainly an area where one should tread with caution.
I would be fine if Christians still only used the TR. I very much enjoy my KJV and have also used a NKJV for many years. For the last 10 years or so I have used the ESV with great profit. If the ESV was based off of the TR I'd still use it. For all practical purposes, they have been the same for me. Let me put it this way: from my perspective there are far more differences in the
way it was translated into English that affect meaning, than in which Greek text underlies it.
And above all, we can be thankful that God has preserved his word with so much certainty! Whether it is just 200 variants (as with the TR) or 3000 variants (with the CT), we can be absolutely certain that the truths contained in it have not changed. Someone said that if all a Christian had were the worst copies of the Greek, he would still believe the same things as one who had the best copies. That is truly marvelous in my eyes.