BTC gambling sites are barely legal as it is, especially in the US.

As for exchanges, the ones that deal with fiat are already subject to KYC/AML regulations, sometimes to the extent of doing long questionnaires about the purpose of the funds that you're trading there (Bitstamp does these sometimes).  Altcoin exchanges could be just one step behind - Bittrex already has a withdrawal limit of 1 BTC equivalent for unverified accounts per day.

If mixing sites closed down there would still be ways to stay private, but the careless and less knowledgeable might get caught.

I also doubt that Bitmixer was under direct threat from authorities but they could feel that they should shut down before it gets too serious.