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.
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.