I had a similar problem on an 850 cc Mini over forty years ago. I thought that it might be the brake shoes snatching but it wasn't that. The wheels were re-balanced but it wasn't that either and the wheel bearings and CVJs were 100% perfect. Braking at low speeds in traffic was OK but braking at high speed caused vibration. Very puzzling indeed.
One day I was driving home from work grinding along through heavy congestion at a snails pace when I dropped something into the passenger side footwell. As the car was travelling so slowly I bent down to pick up the item. As I bent down my eyeline ran along the bonnet and the bonnet seemed to be going up and down!
Once I got home I jacked-up the car and placed a brick next to each front tyre in turn. As the wheel was rotated the gap between the tyre and the brick varied by about 3/8" (10 mm in Metric) I had an oval tyre! This problem had been bugging me for three months and the tyre still had plenty of tread left. Possibly the fact that the tyre was on the side that got blasted with hot air from the radiator had caused the problem.
I did read about a bloke with a Jaguar having a similar problem over a mere one eigth of an inch. As the tyre still had loads of tread left the blokes garage sanded-down the high spot (not sure whether this would be legal today) This cured the Jaguar!
Watch out for oval tyres and only use round ones!
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