Note that log(n) is the length of the prime, so if you are using 2048 bit primes, log⁶(n) is quite large. I don't think anyone actually uses one of the general deterministic primality testing algorithms in cryptographic applications.

we do not have a fast, certain, algorithm for large numbers. The only thing we have for large numbers is “it’s probably a prime. if it’s of a special form, such as messenne, use this other algorithm that takes like 10 days to confirm whether it is actually a prime. If it’s not a number of a special form for which we have specialized algorithms for, which are still pretty slow for large numbers, well you’re SOL, and if you really want to know for certain if it’s a prime or not you better use one of the clever brute force algorithms that do things like only check odds and only check up until the square root of a probable prime, and are still slow as all get out”

So speed improvements are welcome for academic use. Although this doesn’t look like it’s a game changer.

It also doesn't reduce the time to O(1). Each of the ranges is of size O(2^(N/2)) for an N bit prime, so it's really not useful at all.