Can I set up my PC router to use headless Chrome?

What I want to do is use my unix router to give outgoing HTTPS requests from machines on my lan a modern browser finger print.

I mostly use my unix boxes for browsing the web.

I resent having to upgrade and reconfigure my whole OS, say every six months just to get be able to run the latest version of a browser that only a handful of websites require.

I would rather use my old browser and not have to re-invent the wheel and just have my router make it appear like https requests are coming from the latest bedazzle browser.

I am aware some may say this creates security risks but I am confident enough in my firewall and OS and use cases that I am willing to accept this risk.