Does any one of those integrate with Mod Organizer or do I have to download the mod (often also with an annoying wait time) and then point Mod Organizer to it. Do they have an API that enables “a new version is out” notifications, or do I have to hunt everything down manually.

It really wouldn’t be that hard, but none of them cares. Nexus kinda has itself positioned well there as they would not have to support any third-party API endpoints in Vortex, but Vortex isn’t even the popular choice for many games.