Success! Now Cmd+Tab (or Alt+Tab on the keyboard) switches windows inside of the remote controlled Anydesk machine. The solution was to switch the Modifier Keys settings back and download Karabiner Elements, then use that to switch Command and Option (Windows and Alt on the keyboard) instead: Cmd+W would close the Anydesk window, and Cmd+Tab would switch to another application on my laptop). Shortcuts like Cmd+Tab, Cmd+W to close a window, or even Cmd+S to save would all happen on the host machine itself (ie. However in Anydesk, while controlling another macOS machine, it wouldn’t see the Command key anymore. Using the Modifier Keys setting in macOS worked great for just about every app except AnyDesk:

I ended up liking a mechanical keyboard more than I thought, so I wanted to use it with macOS as well. AnyDesk for Mac provides the features and tools you need to connect with desktops or servers in any location.

I just got a PC for the first time in a (very) long time, and got a Ducky One2 keyboard (Windows layout) to go with it. How to use a windows keyboard with Anydesk on macOS

Any desk for mac