The A3 TDi I bought and overhauled for my daughter in June suffered similar. Didn't test drive as it was bought privately during full lockdown and was in the middle of the city so wouldn't have got it above 30mph. Bought with leeway to do work but I was shocked once up to speed on the motorway driving home to feel a dreadful vibration from 60mph. Felt more through floor than steering wheel... as if the driveshafts were bent or the gearbox was about to fall out! Worse I've known in decades of driving and not a typical wheel balance feeling.
Tyres were first check and were actually due for replacement as mixture of Chinese brands and I always fit premium. Both front tyres showed a degree of out of round/tread distortion when jacked up and spun with a fixed point just touching the tread to judge the distortion amount.
So I swapped front to rear as a test and 90% of the vibration went. Then fitted a new set of tyres and smooth as silk.
The amount of tyre distortion seen wasn't as great as the feeling of vibration indicated and you could easily have just balanced the wheels...still had the vibration... and moved onto wheel bearings, driveshafts etc... thinking the tyres OK.
I've seen this quite a bit over the decades and tyre fitters often swear blind they can balance anything out but if it ain't round it might show as balanced in free air on the machine but once back on the road still no good..
Fingers crossed a front to rear swap will prove the point for you... assuming the rears are running true!