Amazon is full of shit. EPUBs only work by using send-to-Kindle which converts it to a file that works (either AZW3 or KFX. Despite the misinformation, EPUBs do not work on Kindle, except if you jailbreak, as you can then use KOReader to read them natively.

That last point is salient, as it means the hardware supports the format just fine. Amazon intentionally does not directly support EPUBs in their software.