Markov chains are super useful in statistics but it isn't obvious at first what problem they solve and how - some further reading that I found helpful:
Note that the point of the markov chain is it's possible to compute relative probabilities between two given points in the posterior even when you don't have a closed form expression for the posterior.
Also, the reason behind separating the proposal distribution and the acceptance probability is that it's a convenient method to make the Markov process stationary, which isn't true in general. (Wikipedia page on MCMC is also useful here).
https://twiecki.io/blog/2015/11/10/mcmc-sampling/
Note that the point of the markov chain is it's possible to compute relative probabilities between two given points in the posterior even when you don't have a closed form expression for the posterior.
Also, the reason behind separating the proposal distribution and the acceptance probability is that it's a convenient method to make the Markov process stationary, which isn't true in general. (Wikipedia page on MCMC is also useful here).