There are lots of examples of improved algorithms, such as your example of factoring, the similarly timed example of chess algorithms, and the much earlier merge sort and simplex algorithms. But in none of these cases did the algorithm completely solve the problem; there are always harder instances that we care about. This is particularly clear with factoring, which is adversarial. (you might count human chess as a solved problem, though)
There are lots of examples of improved algorithms, such as your example of factoring, the similarly timed example of chess algorithms, and the much earlier merge sort and simplex algorithms. But in none of these cases did the algorithm completely solve the problem; there are always harder instances that we care about. This is particularly clear with factoring, which is adversarial. (you might count human chess as a solved problem, though)