I agree about some of those songs, but it is a chicken and egg thing-someone has to pull out the stops to try. Could and might Hamilton? Perhaps; it is groundbreaking in so many ways, it could add this one. But it is one thing to make a push and another to have it succeed. Many try, few do.
I think it because the best songwriters have, for the most part, been unwelcome on Broadway for decades until fairly recently. That's changing and hopefully this situation will fix itself. It is not a quick process, but we are evolving from where things were (Jason Robert Brown, Andrew Lippa, etc) to a point where aging rockers (Lauper, Sting, etc) started writing shows and now we are finally seeing some legit contemporary songwriters (Duncan Sheik, Sara Bareilles) come into these treacherous waters. The next step is to gain respect-how about a song that's actually on the grammys instead of in the "and also receiving a grammy" category? Recuperating from 50+ years of exile doesn't happen overnight.