Central computer with terminals in the rooms

No, you want 3-4 monitors in 4K. No ThinClient can do that!

I would simply use the Dell Wyse 5070 in the maximum configuration - 4x4K + 2xHD possible. Whether that makes sense and how it works, I cannot say. But Dell advertises up to 6 monitors.

thank you for the correction
We have not been working with ThinClients for years. Our customers need "all" FatClients anyway. And in addition, my brain is still running version 2.0b, it was not up to date ;) Nevertheless, he forgot the necessary server in his calculation.

??? All that is possible ???
Just as an example: we have a very tech-savvy colleague here who has installed a FluSi (MS Flight Simulator) in his basement with several FHD projectors and some high-performance PCs, the total costs are currently around 35-40K (but will increase over time).

Are you willing to spend that or more? If yes, okay. I can gladly (just a little) advise you online. If not, the topic is already over here (actually it could have been closed right after the OP).

Hardware always follows the need. That means you describe your needs in detail! (For example, I ask for exact software including version!) Then a tailor-made hardware and software solution is created.
Customers who come with their wishes like you get kicked out immediately! That is basically stealing time for those running this.

Then just kick me out, the way you treat new members, this is no fun here anyway.

Oh , then tell us more about your requirements. Then we might be able to help out for fun.

If you only say more computing power than a laptop, then the normal answer is a regular desktop PC. And if it should be silent, then comes water cooling. That works perfectly silent even with the most expensive (highest performance) CPU while gaming. Personally experienced with my relatives in midsummer. Just a few more radiators and done. Then the water cooling costs about 1000€. That meets all requirements: powerful, silent, four monitors. Super easy. And the room is always nice and warm even without heating. ;)

How old are your kids anyway? They will probably want a gaming PC with colorful LEDs / buy it themselves instead of using dad's tinkering stuff. Or they don't like PCs and a silent all-in-one PC is enough. Or a tablet.

If your opinion is fixed, then search for kvm over ip or similar keywords. But those are all bad crutches, or it gets expensive. Very expensive if you want more than FullHD at 30Hz. And you definitely want that, you describe yourself as tech-savvy. DisplayPort 2 transmits up to 77.37 Gbit/s, by the way. Have fun transmitting that 4x over the network ;)

That only leaves the wall breakthrough, with the computer/server standing directly behind the wall o_O

Why, this is not superzapp's company here.
Here you only fly for links ;)

Seriously:
superzapp offered you free, professional advice. Stay friendly :)

There is a good reason why the combination Mainframe+Terminal eventually disappeared. With multiple computers, you are simply more flexible and each computer can be optimized for what it is supposed to do. You want a jack of all trades, but that does not exist. Instead, put a computer in each room that is optimized for what it is supposed to do and place a fileserver in the technical room. There are also "Clouds für zuhause," just take a look.