Network via TV cable?

I have internet via cable (Vodafone), from the 4 ports of the Fritz! Box (in the basement) I go to a patch panel and from there Cat-6 cables go to some rooms on the ground floor on network sockets, there I either have a GB switch or Fritz! WLAN repeater on and everything works great. So far so good.

After home schooling is again the order of the day, I would now like to connect the laptops of my 3 children in their rooms not via WLAN but via network cables. In every children's room there's already a TV cable that is not used. Since the heroes then inserted the cables into the wall and plastered them, i.e. Not laid them in the empty pipe, I find it difficult to exchange the TV cables for Cat 5 or 6. (If anyone here has an idea, I'm very welcome). The question now is, can I use the TV cable to get a decent network connection via this, if necessary with a media converter, if so which one and what do I have to pay attention to?

Of course there's something https://www.amazon.de/...B00BALOJJG

Should also be reasonably good because Coax is shielded.

But with the prices, I would think twice.