Why do not the broken vases “grow together” back, no matter how hard we press the fragments together?

Due to the impossibility of returning the original attraction to the molecules of the broken thing.
By our own efforts, we purely physically cannot bring molecules closer to each other so that they begin to attract.
Also, the irregularities left at the split site will prevent us from ideally leaning the broken off piece against the original object.
Gaps will always exist, since the residual stresses remaining after the glass / ceramic production are unevenly distributed over the finished product, hence the uneven shape of the fragments.