I would like to know what physicists think about the propensity viewpoint.If this latter one is in line with physics and especially Quantum Mechanics. Otherwise, what is the most coherent philosophical perspective on probability with Quantum Mechanics?
Most physicists don't care about the interpretation of probability. Nor do they care about the controversy over the "interpretation" of quantum mechanics.
The philosophy of probability is not in a good state. It is mostly divided between frequentists who think that probability is defined by long run frequency and Bayesians who think that probability has something or other to do with credence. No real experiment involves measuring an infinite set of systems, so the frequency interpretation is irrelevant to all real experiments. It's also difficult to see why credence should play any role at all in physics.
The propensity interpretation of probability has the merit of saying that a theory of probability has to be relevant to real experiments, but falls down when it comes to explaining that relevance.
There are various attempts to explain probability in quantum mechanics:
but most physicists don't seem to think there is any problem here worth considering. I think this is much to their discredit since having a theory where one of the most important concepts is unexplained makes properly understanding, testing, criticising and replacing that theory impossible. David Deutsch has made some interesting comments on these issues: