In my workplace we use a propietary solution that does a similiar thing.
This device gets an HDMI as imput, encodes it in h264 sends on the network to a multicast IP. Then you can have an arbitrary number of client devices that subscribe to the stream on the multicast IP decode the h264 and output it on an HDMI port.
In the end raspberry pi has a similar hardware than the renders on this setup.
We were thinking about VNC and similar software but in the end those things don't look well when you have to show videos.
One could have a hardware h264 encoding card on a pc and use it to generate the desired stream and publish it on a multicast IP. Then use several raspis to get this stream and decode it.