You mentioned you current stream your TV and are unhappy with the quality. Before tackling the question about network hardware and configuration, I think you need to share what devices you are using, and if you have any way to determine the bandwidth/latency for each of the devices, as well as what speed connection you have with your ISP (marketed vs actual), and how that compares to what you’re getting at each device. Also how many simultaneous streams you are using at once.
From there, you mentioned you have 10 runs of CAT6, do you need all 10 runs to be connected or less? This will help determine if you can use a relatively standard WiFi router (generally has 4 ports, some higher end models have 8), or if you also need to have a network switch.
I’m far from a network SME, but in our house use a wireless mesh system from Asus as well as some network switches to accommodate hardwiring as much of the high bandwidth load as possible (eg, all media streamers/smart TVs, desktop PC).
To my earlier question about testing bandwidth, free apps/sites such as Speedtest.net or Fast.com will provide some visibility to you, and I suggest running the same test a few times at each each device to rule out any anomalies. Also, I am not a fan of ISP provided routers/Wifi Access points, and generally only use the ISP provided modem (gateway) to connect to my router where I can better personalize settings to my needs (eg, network name/password, guest network, monitor firmware upgrades, etc.). Yes, many routers have a wealth of settings and it’s easy to become overwhelmed, but often the out of the box settings will meet most needs.
At one point, I also evaluated the TP-Link Deco mesh system and the set-up for that was really straight forward with a very clean and minimal app that didn’t overwhelm to much in way of technical terms.
Lastly, if you want to use the CAT6 wires, you are going to need to properly terminate them. You should look at a tool/kit similar to the below (I just grabbed the 1st one that came up on Amazon, not a recommendation on the brand), but you need the crimping tool, the connectors, stress relief jackets are plus, as well as the testing tool to ensure all 8 pins on your terminals are crimped and seated correctly and in the proper sequence ensuring passing a solid signal.
Hopefully this didn’t overwhelm you, and I’m sure there are varied opinions and approaches to solve your needs - this was just some through an about how I would tackle the situation.