I would run a few speedtests from a few different devices ookla and the like to get a better picture of if ypu are actually using the faster speeds.
T-mobiles home internet is likely over gsm. I have herd good things about that but I would look at your current provider to see if they have cheeper plans. Before making the switch to t-mobile.
However as much as the other person was downvoted. I would doubt that you would notice much of a difference having a 100mbps internet package with your usage. That $50/ month is pritty significant.
There are passive hdmi to ethernet adptors. I am assuming you can use them in reverse and get ethernet over the cables I would assume all pairs are connected.
We installed one at woek for hdmi but never checked to see if it used all 8 pairs.
https://www.milestek.com/p-16217-hdmi-over-cat5ecat6-extender-balun-passive
This lookes really expensive the set we used was super cheep and had female hdmi for wallplates.