Ethernet wiring was forgotten - wiring?

That's right, my mistake, at 1000Mbit all 8 wires are indeed used.

WLAN can't compare to Ubiquiti UniFi :)

Do you see huge problems and performance limitations?

In real life, no.

So if intense online games are to be played, the kids won’t be thrilled about an additional 5-10ms,

Normal switches forward at most in the double-digit microsecond range. Ping times, on the other hand, are measured in milliseconds (ms).

To get to 5-10 ms, you already have to cascade a large number of switches.

How much money can or do you want to invest? If it's about space, fiber optic would be an option. Otherwise Cat7 Duplex (do not cascade). I would also recommend Ubiquiti unifi. Best to get the switches right away as well. If it's about speed, 10gb switches but that would be rather overkill. For example 3 x US-8-60W or US-8 if no POE is needed on some floors 1 x Cloudkey APs depending on conditions and requirements. As a router a USG That way you would have a complete system that could be easily and centrally managed. Vlans etc. would also be easily possible. If you have no high demands, unmanaged switches (20-40€ each) are also sufficient. For the wifi you naturally need additional ones.

Unfortunately, it was forgotten/saved to pull star-shaped Ethernet cables.

You poor thing, today that's almost like using the toilet across the yard :-(

That's why my idea for the wiring

I would say the makeshift solution is hardly inferior to a planned solution. I don't see any errors.

There are TAE sockets (sometimes even several) with normal telephone cables in every room.

And? - How different are their lengths, can you also count twice four as eight there?

The cables lie in 2 corners, so it will be difficult.. and then the shielding.. I think if the telephone cables are not lying in empty conduits, which I assume, I will implement (have implemented) my makeshift solution. I'm a bit scared of drilling from the basement to the lower ground floor. There is underfloor heating there.