Not an expert, and not answer to this exact question, but the video https://www.youtube.com/watch?v=PtsrAw1LR3E https://www.youtube.com/watch?v=PtsrAw1LR3E and

Not an expert, and not answer to this exact question, but the video https://www.youtube.com/watch?v=PtsrAw1LR3E https://www.youtube.com/watch?v=PtsrAw1LR3E and these slides https://pdfs.semanticscholar.org/41c5/e01cf7e3975efdc5a27f7484cdbba6e63da6.pdf https://pdfs.semanticscholar.org/41c5/e01cf7e3975efdc5a27f74... could answer some of your questions. One way to think about this, without any number theory requirements: if we keep throwing a coin with probability of heads p which is around 1/2 at the beginning, but then keeps decreasing on the rate of log n / n, we would expect (using undergraduate probability theory) that two heads indeed come up very close infinitely many times. As primes up to n have the frequency of log n / n, this would be enough if the primes behaved like coin flip results. However, that is not the case, as primes are leftovers of some sieve, rather than totally independent and random. The goal of some of the current methods is to establish a good ``probabilistic'' model of the primes, so that this intuition can be transformed to reality.