So I tested Remmina some more, and yea, my expectations for the exec plugin were broken. It only runs commands locally, I really thought it was for sending saved commands remotely.
I guess ARD is king, at least from the solutions I know of. Personally, I achieve the “see all screens (or well ssh connections)” and the “send command across all connections” with tmux. And if I need RDP, SPICE or VNC, I use Remmina.
Windows users can use Royal TS, but that is paid and proprietary.