I’ll be emailing the site admin… or some contact at the site, but, is there anything else that can be done to try to put pressure on these websites that tell me “you’re not getting the best experience… download Chrome.”?
I know Firefox has a “Report a broken site” feature, but, the site isn’t technically broken. They’re just telling me to switch browsers.
You could try posting on https://webcompat.com/ to report sites that insist on using another browser or sites broken on your browser.
How this service works? They try to contact the website owner?
It’s a volunteer undertaking. Basically people volunteer to report, triage and communicate the issues. If any bug is confirmed to be a browser bug or a website bug, then some volunteers will create a ticket on the browser or websites support sites.
Again there is no guarantee that all issues will get resolved but it’s a good place to see if the issue is widespread.