Central computer with terminals in the rooms

I would outsource the main computer to the HAR or something like that and build the study room right next to it. This way you can maintain the hard connection via HDMI, USB, etc., and the study room remains quiet.

For the children, etc., I would however also go for NUCs or the mobile variants.

When it comes to media access, you could work with Plex or something like that and access the main computer in the house from all clients.

What data volume is it about?

It is about a few terabytes. But it is also about the fact that I would like to back up this data locally. These are also company data that I simply do not want to put in a foreign cloud.

What performance do you need? What requirements must the computer meet?

Currently I have a PC for about €2000 and another one for €1000. Connected to them are 3 monitors, each 32 inches (to be expanded to 4 monitors).

What should the kids be able to do and why no separate PC? How many users

There are 4 other users. The kids do not need to be able to do anything special. I thought that if I centralize the hardware for my office, it also makes sense to centralize the entire hardware directly.

Then rent a dedicated server and create your own cloud with Nextcloud. What kind of applications? Remote is not good for gaming. Everything else should be well covered. Do the 4 screens connect to one PC or to different ones? So the 4 PCs are for you and three other users or do you use all 4?

This is not about "needing". I find it completely okay. You have to see it as a hobby.
For the amount of data, I would recommend a server + storage space or a very powerful NAS. It just costs a lot of money.
On top of that, one or more VMs. Access them with client PCs via remote.
That way you can have your data locally. A backup on one of their devices/drives, preferably somewhere else, is still necessary. Also a UPS.
I have something like that in my house too (oversized network, a large NAS and an additional one...) ;)
The biggest "problem" is the multi-monitor solution via remote. I know that Igel (dot com) implements something like that with Dell Thinclients.
For your children, one monitor is enough. That is simple.

As long as you don't specify exactly what you are doing with it, it's just groping in the dark.
PC for 2000 and 3 monitors says nothing about the requirements.