This will take you less than 30 minutes with playwright and or puppeteer. A browser extension probably won’t work and would take at least 10x as long to write.
I've tried something similar using Selenium. It worked for a while but then started failing at a site that was using Cloudflare's CAPTCHA. If my browser was launched by Selenium it would end up in a seemingly endless CAPTCHA loop. It failed using both Chromium and Firefox.
I was able to get it working again for a while with some experimental option settings in the Chromium driver which apparently made it harder for a site to figure out that some sort of browser automation was in use, but maybe a couple months later that broke too.
I found a Chromium driver that was specifically made to not be detectable. That didn't help.
I just gave Puppeteer a try, with Chromium and Firefox, and they both got similar CAPTCHA loops.
With Playwright same thing with Chromium and Webkit. With Firefox, to my surprise, it actually worked. In fact I wasn't even shown the CAPTCHA. The checking for a human passed without requiring interaction.
So...looks like Playwright with Firefox might do the trick, but I'm a bit concerned about how reliable that would be long term.
That's one of the reasons I was thinking about an extension--most people using a site are going to have extensions so they aren't going to be able to use the existence of extensions as a sign that a visitor is a bot.
A lot fewer normal users of a site will be using web testing automation frameworks, and so if the site can detect those it might trigger anti-bot measures. I wonder if Cloudflare not being bothered by Playwright in Firefox is due to something Playwright is specifically doing to be undetectable, or just something Cloudflare hasn't gotten around to acting on yet? (I assume it is nothing Firefox specifically did, because of Firefox getting stuck in the CAPTCHA loop under Puppeteer and Selenium).