Within the frequentist "school of thought" how are beliefs updated? Background
Edit: I realize my use of the word "hypothesis" is confusing, I do not mean specifically a null hypothesis. I mean a proposition that something is true.
From my limited understanding, Bayesian probabilities represent beliefs. A scientist may therefore assign a belief/probability to the statement that a hypothesis is true before conducting an experiment or study, and then through formal mathematical reasoning calculate an updated belief as a numerical value (probability) when the results of the study are available.
From a frequentist point of view, a probability is not a belief. Nevertheless, it is common to find phrases along the lines of "Our study strengthens the evidence that H is true". Given that a study has produced results that gives "supporting evidence" to a hypothesis, it seems reasonable that a frequentist would have a "stronger belief" in this hypothesis. Regardless of whether the prior and posterior beliefs are represented by numbers or not (but not probabilities), it undoubtedly seems that there ought to be an order between them, such as "belief after study" > "belief before study". But exactly how this updating of beliefs would happen or how to convey how much more one believe a hypothesis is true after a study compared to before the study, is unclear to me. Granted, I am quite ignorant of statistics.

Question: Within the frequentist school of thought, is there a formal / mathematical procedure for updating beliefs?

If there is no such procedure, it seems difficult to make sense of a scientist saying that a study strengthens the evidence that something is true, beyond a "more than" and "less than" perspective.  The mapping from prior and new data to beliefs seems a lot more opaque to me from the frequentist perspective compared to the Bayesian one. Sure, the Bayesian have subjective priors, but given those priors, the data and chosen analysis it seems very clear exactly how the beliefs are updated through Bayes' rule (although I know frequentists can use Bayes' rule too, just not for beliefs). On the other hand I hardly think someone employing a Bayesian methodology necessarily would actually let an obtained posterior probability represent their exact belief about something, since there can be a lot to doubt, disagree with or improve in a given analysis. I'm not trying to instill any debate between "Bayesian vs. Frequentist", I'm far too ignorant to have an opinion. Hopefully this question is not nonsensical, in that case I apologize.

Disclaimer: I have a Bayesian bias.
The purpose of frequentist hypothesis testing is to reject the null hypothesis: that is not the same as proving the alternative hypothesis. Such experiments don't give you the "evidence that $H$ is true", even if you can hear people making claims like this. The $p$-value is the probability of observing the data $D$ more extreme than observed given that the hypothesis $H$ is true $P(D>d|H)$.
The Bayesian posterior probability is the other way around, the probability that $H$ is true given the data $P(H|D) = P(D|H) P(H) / P(D)$. So in fact in the Bayesian setting, you do update your prior belief $P(H)$ about $H$ given the observed data, while in the frequentist setting you don't. Frequentist experiments don't tell you how likely $H$ is and because of that, it doesn't give you a direct framework for updating your beliefs about it. If you rejected the hypothesis that $H=5$, it still can be anything (just not $5$), and you still don't know what it is.
There is also maximum likelihood, "find $H$ such that the likelihood of observing $D$ is highest under it", but again, it doesn't tell you what $P(H|D)$ is.
Finally, if you consider probabilities to be a measure of beliefs, you are already taking a Bayesian side. Thanks to adopting such a viewpoint, you can measure how true it is given the data. In a frequentist setting, you can only make the "assuming that it is true" claims.
