How about WLC 32?
Question: How is the Grace of God Manifested in the Second Covenant?
Answer:
The grace of God is manifested in the second covenant, in that he freely provideth and offereth to sinners a Mediator,(a) and life and salvation by him;(b) and requiring faith as the condition to interest them in him,(c) promiseth and giveth his Holy Spirit (d) to all his elect, to work in them that faith,(e) with all other saving graces;(f) and to enable them unto all holy obedience,(g) as the evidence of the truth of their faithh and thankfulness to God,(i) and as the way which he hath appointed them to salvation.(j)
a. Gen. 3:15; Isa. 42:6; John 6:27 b. 1 John 5:11–12. c. John 3:16; 1:12. d. Prov. 1:23. e. 2 Cor. 4:13. f. Gal. 5:22–23. g. Ezek 36:27. h. James 2:18, 22. i. 2 Cor. 5:14–15. j. Eph. 2:18.
Or 38:
Question: Why Was It Requisite that the Mediator Should Be God?
Answer:
It was requisite that the Mediator should be God, that he might sustain and keep the human nature from sinking under the infinite wrath of God, and the power of death;(a) give worth and efficacy to his sufferings, obedience, and intercession;(b) and to satisfy God’s justice,(c) procure his favor,(d) purchase a peculiar people,(e) give his Spirit to them,(f) conquer all their enemies,(g) and bring them to everlasting salvation.(h)
a. Acts 2:24–25; Rom. 1:4; 4:25; Heb. 9:14. b. Acts 20:28; Heb. 9:14; 7:25–28. c. Rom 3:24–26. d. Eph. 1:6; Matt. 3:17. e. Tit. 2:13–14. f. Gal. 4:6. g. Luke 1:68–69, 71, 74. h. Heb. 5:8–9; 9:11–15.
Or 43?
Question: How Doth Christ Execute the Office of a Prophet?
Answer:Christ executeth the office of a prophet, in his revealing to the church,(a) in all ages, by his Spirit and word,(b) in divers ways of administration,(c) the whole will of God,(d) in all things concerning their edification and salvation.(e)
a. John 1:18. b. 1 Pet. 1:10–12. c. Heb. 1:1–2. d. John 15:15. e. Acts 20:32; Eph. 4:11–13; John 20:31.
These are only a few Q&A's in the Larger Catechism which show the work of the Spirit in the Covenant of Grace. I would refer you to these, which show the benefits of the Covenant of Grace administered by the Spirit of God. As to your comment of "making too much of Jesus' word concerning the helper coming", I would only respond by stating that I assume I'm among Reformed friends, who have no strange dichotomy about the work of the Spirit in both Testaments, beyond that fuller effusion of the Spirit spoken of in Ezekiel 36, Jeremiah 31, and Hebrews 8. The saints of the OT also enjoyed their blood bought redemption through that same Spirit.
I see no urgency as you do, and instead as I look at the Scriptural revelation I see that our divines have done a good job of representing the Son as the focal point of the Covenant of Grace. Seeing the benefits of His work wrought through the Holy Spirit is consistent with the Biblical record. Note that in Acts 1 the Spirit is called the "promise of the Father". When we look at the New Testament and see that the word "promise" is often put for "covenant" the difficulty you speak of is really not a difficulty at all, but the confessions following that lead, focusing on Christ. In other words, to whom is the Spirit promised? He is not promised to the elect, except in Christ. It is Christ that has secured His blessings to all His elect by the Spirit.
Have a great Lord's Day tomorrow.