I don't think that this is a reason not to have children's music for children's Sunday Schools. Instead, I would make it a priority to include children in the called worship service where they would learn hymns, etc in addition to songs which are easier for them to sing as children. For instance, I would never wish that my son had not learned "Jesus Loves Me" as a 2 year old. Nor would I begrudge him tunes like "Lazarus Was a Wee Little Man" b/c these songs have taught him Bible verses, have underscored Scripture lessons we've taught, etc. That said, a few of the other tunes he learned early were the Doxology, the Gloria Patri, and the Kyrie Eleison, in addition to an ever growing familiarity with general hymnody. (One of his favorites is "Be Thou My Vision," for instance.)
As for skipping VBS... my pastoral advice would be to rethink that. VBS is a great way to get kids excited about church, as well as (if done properly) a way to help them learn evangelism. For instance, at every church I've served we have really emphasized to kids that their playmates who don't go to church could be invited to VBS to learn about Jesus, make new friends, and have fun. It has led to a greater exposure of the Gospel to unchurched families, and has even led some families to attend church & come to faith, not to mention that it has helped kids learn to do evangelism from avery young age.
As for Psalms, don't forget that a lot of the Sunday School tunes (and even some of the happy-clappy tunes) are from portions of the Psalter. (e.g., "This Is The Day," "I Will Enter His Gates With Thanksgiving In My Heart," etc.) Also, others are portions of the NT or OT set to music ("He Has Shown Thee, O Man," "Sing Hosanna," "Rejoice In the Lord Always," "Isn't He Wonderful," etc.)
Don't forget that the Gregorian chants, which were based on OT Temple music, were often short passages of Scripture set to acapella music. Just b/c the tunes are geared toward children does not mean that we can't teach it to our children, or even enjoy it ourselves. After all, it was the Lord Who said, "Let the little children come unto Me... for of such is the Kingdom of Heaven."
Shalom,