何でも generally means “any (one) thing”, not “everything”.
何でもいい → Anything is good.
何でも忘れる → He forgets anything (he gets told).
何でもできる → He can do anything.
全部できる → He can do everything (= all of the things).
全部できた → He was able to do everything.
全部忘れる → He will forget everything.
全部忘れた → He forgot everything.
全部いい → Everything is good.
全部よかった → Everything was good.
The reason 何でも忘れた is weird is for the same reason “He forgot anything” is weird in English: “anything” raises a potential but unspecified thing into the discourse, which doesn’t work well with the past-tense, because you should know what is being referred to. “Everything” raises all the things in the set being discussed, which goes fine with the past-tense.
This is not to say past-tense doesn’t ever work with “any”/何でも; in particular stative predicates work okay: 何でもよかったのに一つも選んでくれなかった “Anything would have been fine but he didn’t pick even one”. 何でもできたのに諦めて引きこもりになってしまった “He could have done anything but he gave up and became a hikikomori.”
This seems to be because the state somehow protects the scope of the quantifier, and doesn’t let it get affected by the past-tense 〜た. I feel like its usage is more common in counterfactuals (like the 〜のに sentences above) though it’s also possible outside of them: 私は(生まれてから)何でもできた. However this feels somehow very slightly odd so I’d recommend against it unless you know what you’re doing.