In principle it comes down to the fact that problem space explodes tensorially with number of electrons e.g. in principle you need grid size of 3N dimensions per every electron so for helium a^6 where a is number of points in the grid (and you probably need at least thousands of tens of thousands to accurately solve differential equations). This can’t be done in practice so other methods (expansion of problem in basis usually) are used.

Unfortunately solving Schrödinger precisely requires incredible amounts of both computing power and theoretical expertise. Qualitative results are up there for most systems thanks to developments in theoretical chemistry but it is still quite manual process that requires expert computational scientist to make sure results are reliable.