I suppose that's possible, but last I looked into it, it was also expensive. If I need $5000 in BTC, how much do I have to spend and how long do I have to wait? Additionally, how many bits of identity do you lose by buying a capable miner?

I'm not sure mining qualifies as "easy", though perhaps not extremely difficult.

For $5000 you can definitely afford a miner. I'm a couple years out of the loop on mining technology and the current & future global hashrate, but with research you could probably get a pretty good estimate on the break even point. An uneducated guess is about a year.

Take a look at Dash (formerly known as darkcoin). It has anonymous mixing built into the protocol. So buying BTC -> dash -> BTC should be able to create anonymous coins. The bits of identity you lose (with this method or buying a miner) only identify you as possible BTC holder, but the actual coins are still untraceable/anonymous.