I have found that both the type of thread used and the type of batting used affect drape a lot.
For example, Warm n Natural batting is considerably stiffer in drape than other cotton battings, undoubtedly because it is needle-punched through scrim. It softens up a little with each washing, but probably will always have a stiffer drape than other cotton battings. Older cotton battings had no permanent bonding in the fibers, were not needlepunched, and had no scrim. Years ago I hand quilted a baby quilt using Mountain Mist premium 100% cotton (what would be considered a vintage type of batting - I do not recommend it because it was difficult to hand quilt!) that was much used over the years. After 10 years of machine washing/drying regularly, I had an opportunity to handle that quilt and could not believe how soft it was. It felt like a cloud! All cotton battings will soften up with washings, but those old style cotton battings soften up a lot more than Warm n Natural would.
I have also found that dense quilting on its own does not necessarily stiffen a quilt, independent of the batting. I once stipple-quilted by machine a small doll quilt using 50wt cotton thread. It came out stiff as a board! Cotton thread is stiffer than polyester thread. If I were to have used a 60wt Bottom Line polyester thread and quilted as densely, the doll quilt would have remained soft. Aside from the looks, I think the 100wt silk thread used by show quilters for dense quilting probably has a similar effect, although I have not handled a quilt like that to know for sure.
I agree that hand quilting, even when dense, allows a quilt to remain soft. Compared to machine quilting, only half as much thread is used. This is probably why cotton thread when used in hand quilting does not stiffen a quilt the same way machine quilting with the same thread would.
In other words, density of quilting is only one of the factors affecting how a quilt feels and drapes. I would guess that batting type and thread type/size play just as much a role.