Does Frequentist statistics still make sense when the experiment is not repeatable? Frequentist statistics is based on the idea that probability should be viewed objectively, that an event's probability is the limit of its relative frequency in many trials, and that probabilities can be found by a repeatable objective process.
Given this, does frequentist statistics still make sense if an experiment cannot be repeated in practice? To give a very simple example. Suppose I have a vase and I want to measure its mass $M$ using a somewhat unreliable scale that gives me readings of $X=M+\epsilon$ where $\epsilon \sim N(0,1)$. I obtain measurements $X_1,X_2,\dots,X_n$ on which I do frequentist inference (calculate the maximum likelihood, confidence intervals of $M$ etc.), all well and good. However, suppose then I accidentally break my vase and the pieces are not retrievable, then is frequentist inference still valid? The whole idea of confidence intervals for example relies on the idea that upon repeated experiments, a 95% confidence interval will capture the true value of the parameter 95% of the time, but if I can only ever do the experiment once, how is the confidence interval meaningful?
The above example is obviously very artificial and stupid but there are countless situations in the real world where repeated experiments or collections of the same datasets is not possible. For example, data that was specific to a particular time in the past, like if you're trying to infer the net CO2 emissions on earth in the year 2005 based on data collected in 2005. In those situations, how does frequentist statistics justify itself? Or should one concede to Bayesian inference when repeat experiments/recollection of data is not possible?

Simple answer, yes the frequentist approach is still viable. We start with a (admittedly idealized) model. Take your vase example. Pre-sample, under the model we know that the confidence procedure with random sampling produces CIs covering the parameter 95% of the time. No actual repetitions are required. Also, if you look at a large number of independent 95% CIs from different problems then if their underlying models were valid (a big ask) approximately 95% would contain their respective parameter.
Regarding your real life example using past one-off data, the question is whether it is reasonable to treat the data as if it was a realization from an underlying model with random variables. For this to give valid frequentist inferences, you would also need to make sure that the choice of model by you was made before you saw the actual data.
