I've asked this question on Emulator Zone, and they didn't really know what I was asking. I hope I'll have better luck here, because most of the games I play were written by members of this forum.
I've been using a Flash app to play these games. As of today, they no longer play. This is not a browser issue. The browser allows me to run Flash, and also to ignore the out of date warning. But Flash itself no longer runs its apps.
The website with the Flash apps is intact, with all the games still accessible from the same hyperlinks, and even the frame in which the game is supposed to play, but it doesn't.
Whatever the exact problem is, it sounds like it must be an extension of support for Flash being permanently discontinued, which happened yesterday I believe.
If you want to play the hacks here, your best bet is to just use an emulator with the patched ROMs. Trying to use a web browser to play them sounds rather insecure, so it's no surprise that you'd run into issues eventually.