I think I can show how probability is not purely in the mind but also an inherent property of things, bear with me.
Lets take an event of seeing snow outside, for simplicity we know that snow is out there 3 month a year in winter, that fact is well tested and repeats each year. That distribution of snowy days is property of the reality. When we go out of bunker after spending there unknown amount of time we assign probability 1/4 to seeing a snow, and that number is function of our uncertainty about the date and our precise knowledge of when snow is out there. 1/4 is a precise description of reality if our scope is not just one day but a whole year. In this case we have a precise map, and our uncertainty is lack of knowledge of our place on the map. What we also know that if we do not have a date or season there is no better prediction and this is a property of things too.
Additionally having probability distribution you can perfectly predict accumulated effect of series of events, and this ability to predict something precisely is an indication that you grasped something about reality.
Returning to the coin 0.5 prediction of one throw is function of our uncertainty, but our prediction of sum of long series where 1 is heads and 0 is tails is a result of our knowledge of coin properties that is expressed as probability
This example needs to be refined, two experiments do not produce the same data, second one has different probability space and additional data point - stopping position, and computing probabilities you should also condition on that stopping point N, fact that this N is screened by other data is nontrivial and waiving it just on assumption of beauty could lead to mistake.
It turned out that in this case that is correct move, but could be a mistake quite easily.
Probability given data is an objective thing too. But point I make is that probability you assign is a mix of objective and subjective, your exact data is subjective thing, distribution is objective, and probability is a function of both.