Just because this could be the root of misunderstandings: My response should specifically not address weights used to obtain unbiased estimation of opinion- or political surveys/polls, but weights used to obtain less biased estimates of causal effects. It is my understanding that these literatures are related but do address different questions.
I did not think of bias being connected to point estimators. I also do not remember discussions of selection bias in the context of (causal) effect estimation, which is what I often have to be concerned with, being tied to either a Bayesian or frequentist framework, but I may be missing something there. When I wrote about trying to obtain an (un- or better less biased) estimate for the a causal effect, I meant obtaining a posterior distribution for a causal effect.
I am rather pragmatic and think that there are two problems to solve: (a) What approach do can I use to reduce bias, this can involve choosing between MRP and weighting. (b) Estimating model parameters. I say that I am pragmatic because I treat these problems independently and did not spend much time on reflecting if this is philosophically sound.
Formulated like this, I’d say that the analyses I do might not be conditional on the data. But I am not sure if what you write accurately describes what I am doing. Specifically, I do not use an observed weight w_i, but I use the data to estimate a posterior distribution of w_i, which I then use in the next step of the analysis.