Your understanding is incomplete. Bayesian statistics are not subjective at all. Instead they objectively describe the correlation between prior belief and conclusion. People may draw different conclusions for subjective reasons, but that subjectivity is in the people, not the statistics.

Now it seems you are objecting to the fact that your initial state of mind affects your conclusion. But it is unavoidable that there is no way to make sense of observation except through the lens of prior beliefs. You can be explicit about it and draw conclusions with a correct methodology about it as in Bayesian statistics. You can be implicit about it and draw conclusions with an incorrect methodology as in frequentist statistics. (See my other post in this thread for an explanation of why frequentist methodology is incorrect.)

Let me offer a simple example. Suppose a pregnant woman you know gives birth to a boy. Is that evidence that babies are more likely to be boys than girls? Obviously it is. Is it strong evidence? Obviously not. Should we upon observing that conclude that boys are more likely than girls? Obviously not.

Now suppose that you have no prior knowledge other than we see about similar numbers of boys and girls. I know that of the last 100,000 babies born in the USA, 51,157 are boys. Suppose we are both told that at the local hospital, 92 of the last 200 babies born were boys. I submit that we both will and should wind up with different conclusions about the relative likelyhood of boys and girls. Why? Because different prior knowledge lead to different prior beliefs, and those prior beliefs when modified by identical evidence lead to different conclusions.