Front drivers' side tire wobbling?

You're describing a broken belt in the tire. Common causes are a puncture that was patched on the inside, leaving the hole open for water and salt to get in and corrode the steel belt, a hard hit or bruise on the sidewall, and just a normal occurrence from age. My experience has been Goodyear tires in particular develop broken belts right about the time the tread is worn to the point the tire should be replaced anyway. Years ago, it was also really common to see broken belts on Uniroyal tires on GM cars. I don't know if that is still the case today. In the late '80s it was so common; we just assumed every GM car that came in for tires had one or two with broken belts.

Here's links to some related articles that may be of value:

https://www.2carpros.com/articles/steering-wheel-shakes-when-accelerating-or-braking

https://www.2carpros.com/articles/how-car-tires-work

https://www.2carpros.com/articles/wheel-removal-and-re-installation

There are two ways to see a broken belt, but you already described one of them. Be really careful if you run your hand over the bulge as they are usually accompanied by wires from the steel belts sticking out. Those ends are really sharp and smart when you get poked by them. When you rotate the tire, you'll see the tread squirm from side to side as it goes around, or there will be a "tumor" or hump on one side, in one area.

A broken tire belt can also occur that presents a much more elusive diagnosis. That is when it breaks over such a long time, the bulge occurs gradually and the raised tread wears down as you drive. With this type of break, the tread that contacts the road surface looks normal, but if you look down into the grooves of the tread as you rotate the tire, you'll see a very small fall and rise of the carcass. That is due to a weak spot. That weak area doesn't provide the support, and lets the axle fall slightly when it comes around. That will cause a thumping feeling at higher speeds. At parking lot speeds, you will typically see the steering wheel oscillate back and forth once per tire revolution.

This photo is the best example I could find of a typical broken belt. See the bulge in the lower left. Most other photos show much worse examples. Most real broken belts aren't always so easy to spot.