An update. Still searching for the source of the problem.
But so far I established:
Most likely it's the booster or the master cylinder.
If you disconnect the vacuum line from the booster the brakes are "released". So, I had thought it's the booster failure but then decided to do a check. I disconnected the vacuum line and took the car for a 6 miles ride on a free road with some 50-0 hard stops . The issue appears again. Not as fast nor as bad as with the booster connected, but it's definitely progressing while driving.

I also tried to diagnose the booster by the way the JTIS says and nothing wrong with it according to the booster test.



Any ideas or thoughts? Master cylinder was clogged when I pushing the pistons back?