This doesn't strike me as a good example of reproducibility in this context. Let me offer another one. Let's go back to your example of the 66% weighted coin. Given the physical properties of the coin, different people could independently come to the same conclusion that the probability of heads is 66%. I would describe this as an "objective" probability, as it's a nice representation of the available information, independently reproducible by different people, given the same information. It's different than "Bob arbitrarily decides that any yes/no question has 50/50 probability", which is inherently subjective.
No two people ever have the same set of information. And very few cases are even as clean as the coin case.
A more typical example is using polls to predict elections. 538's model ended with Biden around 90% to win. Andrew Gelman's model at the Economist ended with Biden around 95%. Do either of those represent objective probabilities?
I would say no, and I think that just because two people happen to agree on a number in a particular case doesn't make it objective. If you want to use the word objective, I don't have any particular objection. I'm not here to fight over words, and none of these words are really well defined enough to be worth fighting over. I don't think it's useful to think of probabilities as "real" in any sense.