Follow up:
I took the X in to Tesla Sunnyvale to get aligned. Turns out there was a failed bushing which explains the handling issues. They ended up replacing the “FR SUSP AFT LINK ASSY” and then performed an alignment (all covered by warranty).
The car tracks a lot better at freeway speeds now. Clearly, the new tires had nothing to do with the previous problems. AP still struggles at 70+ mph, though, wobbling back and forth in the lane, making it unpleasant to use. This is in direct contrast to our 3, which AP does a great job with. Upon my post-repair feedback about AP, the service center recommended another service request, which they would use to initiate an OTA AP calibration. Hopefully that helps.