It seems that there are many versions of this question that have answers when I look it up. But it seems quite clear that LUS should find a factor within an asymptotically quadratic transformation of the optimal run-time for factorization. So then it seems obvious to use the AKS primality test in P to efficiently determine the existence of some non-trivial factor and if it exists then we should be able to use Levin's Search, right? So that is essentially an efficient algorithm in theory for integer factorization? Why is this not widely known and taught then? So I'm probably mistaken and so I want to know why this is not theoretically valid. Since the AKS test is valid, why isn't LUS?
In a practical sense, Levin search is not useful. It has an enormous constant factor (exponential in the length/size of the optimal factoring algorithm). This makes it of little use in practice.
In a theoretical sense, it is interesting but of little impact, because for theoretical reasons our primary interest is normally about what algorithms exist or not, and Levin search gives us no insight into that.
I'm not sure where you got the notion of a quadratic slowdown. I don't believe that to be accurate. See, e.g., http://www.scholarpedia.org/article/Universal_search, https://cs.stackexchange.com/q/57179/755