There’s one huge difference between batsmen who don’t walk, and the fielding side claiming a false catch or stumping... the fielding side is the one who appeals - they are the “aggressor” (the ones proactively seeking a dismissal). If, in a bizarre world, the batsman got to say “not out” or “out” after every ball, then perhaps you could equate the two. The fielder can proactively tell the umpire that the catch is not clean...the batsman can’t tell the umpire that he didn’t edge the ball (for a caught behind appeal) or that there was an inside edge onto the pad (lbw appeal).
So, because the fielding side is the aggressor (ie they alone have the power to initiate an appeal and communicate with the umpire), it is totally unacceptable and inappropriate for them to claim a false catch / stumping, and totally acceptable and appropriate for the batsman to stand his/her ground until the umpire gives them out.