Personally I’d rather have multiple connections to the router. If that’s a shelf in front of this setup, I’d be tempted to put the router there and give each switch it’s own connection directly to the router. With the switches currently there you’d replace the blue cable connecting them (port 8 on left, port 1 on right) with a cable from each to the router. It splits the maximum possible devices that would be competing for bandwidth to the router.

Of course if that’s a terrible place for the wifi then I’d do what you said.

What you’re looking for is a MoCA network. It won’t be cheap, but it is possible. You’ll need an adapter in every room you want wired internet, and high quality splitters. Setting up a Powerline network might be cheaper. But quality of that will be dependent on the quality of your electrical system.