Maybe someone has brought this up, but Westminster, Larger Catechism 156 has this to say about private reading of the Scriptures and uses some of the proof texts discussed already in this thread:
WLC Q. 156. Is the Word of God to be read by all?A. Although all are not to be permitted to read the Word publicly to the congregation, (u) yet all sorts of people are bound to read it apart by themselves, (w) and with their families; (x) to which end the holy Scriptures are to be translated out of the original into vulgar languages. (y)
u DEU 31:9, 11-13; NEH 8:2-3; NEH 9:3-5
w DEU 17:19; REV 1:3; JOH 5:39; ISA 34:16
x DEU 6:6-9; GEN 18:17, 19; PSA 78:5-7
y 1CO 14:6, 9, 11-12, 15-16, 24, 27-28
Also, the Kirk of Scotland has the Directions for Family Worship which indicate Scripture reading is a part of private and family devotions; but it does not include any scripture proofs.