3 bar on the rear tyre?

And you complain about the traction control telltale light flashing?
Even the KTM with 4 suspension modes was the same, an equal disaster in every mode - just different versions of horrible - the handling was the same basic nastiness
And you're sure you're not a smidge biased? Or all the bikes you ride are an utter engineering failure?
I don't consider myself that experienced or fast by any standard, so I would normally address any issue I have on the bike starting with:
1. Me. Am I inducing this behaviour through a riding fault of my own?
2. Are the shocks/tyres in tip top conditions? Following that setup?
3. Surface issues/mechanical (something broken) problems.
Seriously uncommon for modern bikes to have blaring engineering problems like what you describe. Unless we want an excuse to ignore errors that are due to ourselves mostly.
I've ridden my hexhead on track a few times for fun and, again, I'm not fast but not exactly bimbling along either, never noticed any of the issues you mention (flex) neither did people that ride significantly faster than me (and have extensive experience) on similar hexhead, non adventure, on sport road tyres.
Main issues I have had was, possibly, stability over hard-ish braking. Overheated tyres and (later) brakes sent the bike around like a boat, but that would have been expected.
I experienced some lack in faith on the front end the last time I had the GS at Brands. I was more than my usual over-caution (I have a safe zone

) when cornering. Later on, while traveling to Italy this became a
very unstable bike on rapid direction changes at speed. I still remember a quick left/right across a sudden dip, coming down the Alps, that almost required a change of pants. Shocks were fine and relatively recently serviced (I have Ohlins TTX on that bike). When in Rome had it checked and turned out to be the ball joint on the telelever, worn out and out of shape, so the front suspension movement in compression and extension wouldn't be smooth. It would compress, reach a point where resisting compression, and then give up all at once.
As you are talking about older hexhead/twincam hexhead bikes, I'd humbly suggest to give it a look maybe as well.
Hope you find the solution!