Why does my steering wheel wobble?

I agree that it could still be the tires. And I agree that moving the tires to the rear, preferably one at a time, is a good way to find out.

If the culprit is identified and is, in fact, a tire, a “road force balancing” will determine whether it can be saved by “indexing” or there’s an internal defect. “Road force balancing” is a process by where the wheel is spun with road force simulated by a spinning drum pressed against the tread. It can detect imbalances and internal defects that regular machines cannot. “Indexing” is a process of compensating for a heavy spot on the tire (no tire is perfect) by placing that spot opposite the heavy spot on the rim. It isn’t normally practiced anymore, but can save an imperfect tire.

Chances are that the “tire guy” just checked the tread for erratic wear patterns and evidence of serious internal failure (like a bubble in the sidewall).