"Did your steering wheel center itself just by replacing the wheel?!?! Amazing."

Yes, and I have experienced this before. This is one reason why swapping front/rear can be a very good diagnostic.

If the rear suspension has not been damaged or misadjusted, it will typically hold its settings a long time, and tires tend to wear evenly. Front tires are subject to a lot more abuse from just normal driving, and because geometry must change for steering input. It is much easier for the front suspension/steering to be misaligned. Developed uneven wear patterns can cause a tire to 'pull' to one side, requiring a compensating input from the steering wheel.

Swapping the tires removes the need for that compensating force, so the steering wheel is now in a different position to drive straight. The less-than-perfect tires now in the rear will have little affect on directionality, and may wear back to a more normal pattern.

Of course, I should still get an alignment, or the 'new' front tires may eventually wear unevenly too. Good luck with the ball joints!

Steve