Has there been "no progress" on classical prime factorization? What about the AKS primality test, a polynomial-time algorithm to test the primality of a number, published in 2002? (This is not my field of expertise; I'm genuinely curious if there's a good reason to discount this as progress towards efficient prime factorization)
Primality testing was essentially solved in the 70s with Miller-Rabin. AKS made that (randomized) algorithm deterministic, albeit at much higher (polynomial) running-time.
For your overall question, the current record-holders for integer factorization wrote a paper on this a few years ago that is probably a good reference
https://hal.science/hal-03691141/file/cryptography.pdf
The (rough) outline of the paper is that
1. theoretically there's been no progress on factoring in ~30 years
2. practically, there have been both improved hardware + efficient implementations driving the progress. They estimate that current nation-states can (classically) break RSA-1024. The cost would be approximately 500,000 core-years of computation. At current cloud prices this is doable on aws for < $1B.
3. attacks against factoring use a technique ("index calculus") that can also be used to attack finite-field discrete logarithm. There were significant advances on that problem in the 2010s (at least for certain parameters, namely the "small characteristic" setting). An easy way to communicate this is that the RSA factoring record is ~830 bits, while the binary-field discrete logarithm record is > 30,000 bits. These significant advances have not been able to be ported over to factoring, nor have they been ported over to medium/large-characteristic discrete logarithm. It is a (very upsettingly) large open question of whether similar-magnitude improvements are possible more generally for index calculus algorithms.
> Has there been "no progress" on classical prime factorization?
Not recently. The primality tests don't really help all that much. We already had polynomial tests that are really fast since the 70-s.
Think about this idea: the output of the counting function for the number of primes ("Euler's totient function") lies almost on the logarithmic curve, and we can compute logarithms quickly to any precision. So we can easily find the general area of the curve that should contain the current prime. And then we can quickly test if the given number is in fact the prime number within it.
This is probabilistic because the prime distribution is not _strictly_ logarithmic. We can imagine that by computing a logarithm we might end up in the next "bucket" and check for the wrong prime.
The fascinating part is that zeroes of the Riemann zeta function encode these corrections on top of the logarithmic curve. If the Riemann hypothesis is correct, then these corrections are _bounded_ and we simply can not end up in a different "bucket" by accident.