Does proving a problem is in P typically mean it will eventually be practically solvable?
One example in which the running time became significantly better but still not necessarily practical is submodular function minimization, which went down from $O(n^7)$ (Grötschel, Lovász and Schrijver, 1981) to $\tilde{O}(n^4)$ (Lee, Sidford and Wong, 2015). The latter paper (and the ones preceding it) has many such results.