I found an Infiniti QX4 I might be interested in as a backup/ outdoor vehicle. 4wd, auto transmission, and 160k mile for 3 grand. No engine lights, although the owner mentioned it has a "slight left and right shakiness" when accelerating and releasing the pedal, and the owner believes it's a rear control arm bushing. Does this sound like a good buy?
the owner believes it's a rear control arm bushing
A left & right shake sounds much more serious than a dirt cheap bushing. Odds are the owner would’ve fixed it too -
As it is, Scotty doesn’t recommend buying old high mileage Infiniti vehicles, let alone one with some potential drivetrain/chassis damage. I recommend you steer clear of this one.

No.
I have a 03 Infiniti QX4. Bought with 150K and now at 200K. Lots of snow driving and off-road driving. It’s a great rig, but expect to put at least 3-6K into it if you are planning on keeping it alive to 300K. I upgraded to OME shocks, struts and coils.
If it’s the left to right shakiness when braking then you need to replace the brakes and set the drum brakes in the rear to do more work than the front disc brakes. My mechanic did that to mine and it doesn’t shake when braking. Hasn’t done it for the last 45K miles.
Check all the bushings of the lower, upper control arms in the back as well as the pan hard bar (not sure if that’s how to spell it).