According to your criteria, each prefix independently has the same probability of being in one place or another, and you are right (independently the probabilities are the same). However, you only take that as a single truth, which is where I disagree. If you debated between joint probability and compound probability, you would have a bit more logic.

because, joint probability refers to the probability that two or more events occur simultaneously. So, it could be interpreted as all the hashes being "written in stone", therefore you could argue that all independent prefixes were generated simultaneously for Bitcoin.

In the case of compound probability, on the other hand, which refers to the probability that two or more independent events occur in sequence, you could also apply it to Bitcoin by calculating the probability that a prefix repeats as you move from one prefix to another.

But in any case, it cannot be taken as a single independent event, it is not like you are looking for a different prefix in each event. In the same way, this book explains it with examples.

"A First Course in Probability" by Sheldon Ross