Can't pinpoint the reason.
I've experienced wobbling before, but that time I was sure speed was the reason. Riding 240+ km/h on H-rated tyres (not to mention one was radial, and another one was bias) was an idiotic idea. Strangely enough, I was able to stabilise the bike just by the book. Hands relaxed, open throttle. Almost soiled myself though.
This time the tyres were new. Avon Cobra WW. OEM sized. V speed rating.
Right after I've fitted these tyres I've been experiencing front wheel wobbling at low speeds - 50-60 km/h even though both wheels were balanced. I suspect that the front tyre was faulty. Changed front hub bearings just to make sure the weren't the reason. Front axle was perfectly seated. Hoped the rubber will break in. Took a 2000+ km trip to Romania with my wife. Didn't help. Had no problems on speeds over 60 though.
I don't thing it was just the tyre. The road looked perfect, yet I remember three sequential smooth bumps, almost unnoticeable, and a longitudinal crack after them. I haven't noticed that crack when I was riding. Saw it later on when watching records from on-bike cams.
I guess it was all together - tyre, road conditions and speed. Thank you all for sympathy. Makes the pain less harsh.