"で" in どこでも is not the particle "で" meaning "at" or "in". Here, "でも" is something like English "-ever" or "any-" in whatever, anywhere, etc.
- だれでも whoever
- だれにでも勝てる I can defeat anyone. (~に勝つ = defeat ~)
- いつでも whenever
- いつまででも (lit. until whenever) forever
- なんでも whatever
- どこでも (in/at) wherever
- どこにでも行く go (to) wherever
- どこへでも行く go (to) wherever
- どこまででも行く go (to) wherever
- どこからでもやって来る come from anywhere
- AとBのどちらでも使える Both A and B are usable.
This で is dropped in negative sentences:
- ここにはなにもない There is nothing here. (*なんでも is incorrect)
- だれもいない There is nobody. (*だれでも is incorrect)
- だれにも勝てない I can defeat no one. (*だれにでも is incorrect)
- いつも17時には帰れない I can never leave my office at 17. (*いつでも is incorrect)
- いつまでも子供のままではいられない You can't stay a kid forever. (*いつまででも is incorrect)
- どこからも電話がかかってこない I get phone calls from nobody. (*どこからでも is incorrect)
- AとBのどちらも使えない Both A and B are unusable. (* どちらでも is incorrect)
According to this rule, "どこでも銭湯があります", "どこにでも銭湯があります", and "どこにも銭湯がありません" are all correct, while "どこにでも銭湯がありません" is incorrect.
There are always exceptions. The following sentences are correct, because it's semantically positive:
- いつでも構わない Any time is fine.
- どこでも問題ない Anywhere is fine.
Can we drop で in positive sentences? It depends.
- *だれも出来る is incorrect (だれでも出来る is OK)
- *なにも食べる is incorrect (なんでも食べる is OK)
- いつも遊んでいる (いつでも is also OK)
- どこも休日は混雑している (どこでも is also OK)
- 一緒にどこまでも行こう (どこまででも is also OK)
As for 「どこにでも」, I feel it's acceptable to drop で in positive sentences.
In conclusion, 「昔はどこにも銭湯があった」「昔はどこにでも銭湯があった」「昔はどこでも銭湯があった」 are all OK as long as they're positive sentences. I think there is no considerable nuances between the three.