"évidence" the noun is just a shorthand for "obvious thing" (most typical usage is « C’est l’évidence même » = “It’s obvious”. « Ce n’est pas la peine d’asséner de telles évidences » = “Such obvious things are not worth stating”).

So, it doesn't look like "obvious" & "self-evident" work as idiomatic English translations of "évidence", but I think hwold's correct to indicate that "évidence" doesn't mean "evidence".

It feels a bit odd to offer that judgement because I'm not French and I suspect hwold is (or is at least a native Francophone), so they probably know this stuff better than I. But I can use dictionaries as well as anyone can: Wiktionary says "évidence" is a noun meaning "obviousness" or "clearness", and Linguee translates it as "obviousness" or "blatancy". Coming at it from the other angle, my old French dictionary suggests "preuve(s)", "témoignage" & "signes" as possible translations of "evidence", depending on the context, but not "évidence". (The same dictionary doesn't even offer a translation of "évidence", consistent with it having a more obscure meaning than simply "evidence".)