Ok, a second update where I got things working.
I uninstalled Mono from my computer, and made sure that the only version of .NET Core SDK 3.1 was the x64 version. If the incorrect version was installed, you need to make sure that it's not still in your PATH either.
(Right click Windows Start Button > System > Advanced system settings (on the right) > Advanced (Tab) > "Environment Variables" button > Click "Path" in the bottom "System Variables" window > "Edit..." button > Scroll down to bottom and you should see "C:\Program Files\dotnet\")
If that looks good, then you need to go to one of the example projects and open up a scene, for instance "3D_IK_Dino_FABRIK.tscn" and add a new completely blank C# script to the root node. Since I no longer have Mono installed I went to Editor > Editor Settings... > Mono > Builds > Build Tool and changed it to "dotnet CLI".
I clicked "Play Scene" and it started just fine. I was then able to go back and enable the Plugins.
Edit: Actually, I'm still having trouble running the "Development_Example_Project" scenes. They can build fine, but I get this error: https://github.com/godotengine/godot/issues/6393