Bitmex doesn't (yet) force you to verify yourself like regular/spot exchanges do. If for whatever reason you trigger Binance's security system and they want your documents, you're pretty much ruined it for yourself. Providing them your US documents will point out that you violated their terms, and who knows what the consequences of that are.

In other words, you should care. Binance is busy trying to market itself as a trustworthy brand where they slowly but surely try to comply with the jurisdictions they operate in.