I used to try and play with the right hand thumb free - with tight enough straps or with my accordion held in its trusty stand it really does seem that you really don't need to push on the box, you can shake the accordion around handy enough. Yet you really can't play the thing that way - having the thumb on the side contributes something else; my guess is stability for the fingers, and that little bit of pressure is a side benefit. With the thumb free I just can't hit the buttons with any accuracy.
Now, is it the fact that the buttons are so small that makes this an issue? Are CBA buttons larger? Or is it the unisonoric aspect of that instrument and the larger size of the case that means the thumb can be free? But there are learner instruments of CBA and PA that are smaller than the largest bisonoric button boxes, and people play those with the thumb free, no problem. Perhaps an oversized case for a diatonic button accordion would be an interesting experiment. Some players of the Steirisch harmonika keep the thumb free, too - those are massive boxes, 4 rows usually, and those big Helikon basses. Is it because of the chordal nature of the music they play that they don't need to have the thumb on the side? Although I seem to also remember reading that some of those boxes are actually unisonoric, too.