Feels like lumpy tires

Firstly, please forgive me if I am novice. I didn’t drive until I was in my mid thirties and this is my first car.

The car listed above is the GLS model. I purchased it used about three, maybe four years ago from a garage, probably with too many miles. But at that time, there wasn’t a rust spot on it, it was a super smooth ride, with a lot of get up and go, and great stopping power. Not to mention fuel efficiency (which it, for the most part, still has).

I don’t drive it everyday (probably two to three times a week) but I have cared for it the best I can. I purchased new tires, and every spring and fall snow and all seasons are swapped and tires rotated. All except one time, the oil gets changed regularly if not before it is due. I brought it in to the actual Hyundai dealership for two recalls; brake light and control arm, to which I believe they repaired brake light? Switch? And did a kind of corrosion preventative spray on the underside of the car.

The problem is that about a year to year and a half into owning this car, it developed a shimmy or wobble like I have lumps on my tires, like when one gets a lump of snow stuck on a tire. The slower I go, the slower the wobble. I am scared to drive it on the highway. Like I said, I bought not only new all season, but also snow tires. I had them balanced and aligned. I have brought it to three mechanics and the Hyundai dealership. They all said they couldn’t feel it, but eventually it effects my brakes so it doesn’t brake smoothly, more like b-b-b–b—b----b----b until it stops.

Few of my passengers have said they could feel it a little.

The tires don’t have any strange wear pattern, cupping or bulging, however I don’t drive much (and this is reinforced by this wobble). My father did a naked-eye run-out test on all four wheels and found nothing. He also did other test that involves trying to yolk the wheels/tires back and forth by hand and he found nothing.

But its starting to effect my brakes again. B-b-b–b—b----bb until it stops.

Is there anything I should be grilling my mechanic about? Ask dear ol’dad to look into? Perhaps with some advisement and luck, I could look into?

Full list repairs, may or may not be pertinent:
Ties, tire balance, wheel adjustment, license plate illuminator light, brakes (twice), brake line, (brakes just aren’t the same anymore), fuel line, new battery, interior fuse box. Missed one oil change, got dummy light, changed oil, light went away.