By "completely unreliable" do you mean it won't start?

Since you're in NYC, do you happen to have an aftermarket alarm system in the car? That would explain a current drain but not the clock.

I'd consider taking the car to another shop or to a dealer.

If your mechanic thinks it's a short in the radio, why not just pull the radio out and see if the problem is solved, rather than replace it? If it's not solved then it can't be the radio, so put it back in and keep looking!

A has been pointed out Volvo batteries don't last very long but if you just replace it without fixing the underlying problem that won't solve your problem.

BTW "low mileage" isn't necessarily a good thing.