Short answer yes, you can.
Long answer
There are a lot of ways to do it, but a good box (STB), with a powerful processor is a must. Make sure you have at least 2 dvb-t2 tuners, 3 or 4 if you can. Consider freesat as well. Most STBs come with DVB-S2, so setting up a small dish will give you more choice, and sometimes a better picture.
Don't skimp on the hard disc. Go for an internal drive inside the STB. Of course USB drives do work, but fail at the most inconvient times (cables are often the problem).
Set the STB up with a static IP address (either with a DHCP reservation, or change your DHCP settings to a smaller subnet e.g. 192.168.1.128-192.168.1.254, set you box to an address outside DHCP range)
You'll need a hard wired gigabit network just like you are planning, but use a decent gigabit switch (Netgear or D-Link are fine). Don't depend on your ISP's router, instead hang the router off a port on the switch. How ever many ports you think you need, double it. So if you've got a STB 2 clients and a router (4 ports) get an 8 port switch. If your count goes to 5 or 6 devices, go for a 12 or 16 port switch. As Abu said, at least 2 network cables to each room. They don't need to be connected, but you'll always find a need for them in the future.
Abu was correct in saying that modern TVs allow you to pick up files on shares, but they are usually DNLA shares. Also each TV manufacturer implements DLNA differently, and some have limits to how many files they'll show. It's an OK solution, but the media players are usually pants to operate.
If you go for smaller client STBs on the other TVs, stick to the same make as you main STB.
If you don't have a specific need for STBs at the smaller TVs, Kodi on a Raspberry Pi 3 is a good, cheap choice (about £55 for a Pi, case, PSU and uSD card). The Pi 4 can get a bit hot without a heatsink, but it is being worked on. It is however very fast.
Look at installing OSMC or LibreElec on the Pi rather than vanilla Kodi. They are good fast boots, and you don't need to mess with the pi settings.
You'll need to download the relevant PVR client from within Kodi, and configure it to point at the STB.
Once the PVR client is enabled, Kodi will add a new section called TV, and all your channels, EPG and recordings from the STB will show up on it.
Enable HDMI CEC (on the TVs) and the TV remotes will controll the STBs / Kodi directly.
You may want to look at a NAS if you build up a big library of recordings, or already have a collection of media, (don't forget music and photos).
Its a lot of stuff to go at, and you'll need to understand each bit to get it all working without hastle. But thats what forums are for. The very fact the you know to install CAT-6 is a good sign.