Most Broadway shows are recorded for posterity (and a lot of non-Broadway shows in NYC as well), but typically they are only available for viewing on-site at the New York Public Library's Performing Arts Branch in Lincoln Center (and you need a NYPL library card) and also they aren't filmed as dynamically as Hamilton was. The big issue is that the producers of many Broadway shows don't want to pay what Hamilton paid for a professional recording and also don't want to cannibalize the ticket sales of the show on Broadway and in tours. Most shows still stand to make a lot more from ticket sales than they ever could with selling the film rights (and remember, Hamilton's rights were sold with the understanding that it would be released in theaters instead of streaming). Hamilton is an outlier because it still makes a ton of money in ticket sales and was able to sell the film rights for a lot more than most shows would be able to. Another problem is that a lot of big shows (Wicked comes to mind) have movie deals in the works (albeit not filmed live production deals) that might limit the ability to sell filmed versions of the live production to content providers.
Basically, Hamilton on Disney+ was a huge experiment that had never really been attempted on that scale (aside from like a PBS recording or two). I think a lot of producers going forward are going to see the value in what Hamilton did and we will see more shows in the future spend the money to make a full-scale recording, but I don't think that's happening until Broadway reopens and producers have the ability to do so, and even then they will still hold off on releasing it while the show is still running and a hot ticket. It is also possible that Disney+ and other content providers might start going in and seeing about buying rights to recordings of older productions that have closed/are no longer on tour, but I wouldn't expect that for anything that is still running on Broadway or actively touring right now any time soon.