My opinion, from a similar situation, use the coax to pull cat6 through and replace it. Assuming it's not stapled down or too tight. In a previous home I rented I didn't want to run cable knowing I would only be there a few years so I picked up MOCA boxes; lets you run eth over coax. I connected a switch in the bedroom for a steam box and smart TV to the router downstairs using this and worked fairly well but never did intensive bandwidth tests. Might be worth looking at.

This isn't a meme, my aunt made this for a family gathering over 25 years ago, into a loaf looking pate with the pickles on top. Didn't tell anyone what it was. Nobody was able to guess but everybody loved it

Is it new? Has it always done this? Have you used any additional wiring harness included? Am not an expert but want to try to help. I experienced this years ago after installing an Ecobee without the additional wiring harness which provides power for the ecobee to run correctly. Each time the furnace would change states the ecobee would reboot. Never happened when using the AC. Something about a "c wire" missing from older houses and needing the additional wiring kit.