Then add a "we work best in chrome" to your error pages as a workaround when it actually fails, not before you even try.

There's no need for the "go use X browser right now" messages unless you're taking advantage of some bleeding edge API that only exists in one place.

It really isn’t for most websites. There are a handful of APIs to be careful about but a mainstream website shouldn’t take much time at all. I do most of my development in Firefox and most of the time it works unmodified in Chrome and Safari. Mobile browsers can be trickier but most of that is adjusting for the form factor rather than the browser.