One browser is not enough. Image formats have massive network effects — people want to share, save and remix images. You need support the format in image viewers, editors, file browsers, mobile apps, websites.
As an exercise, try having your avatar only in the WebP format and see how useful it'll be.
WebP is mostly handled via transparent proxies - meaning people with proper browsers and Android mobile devices get WebP (large market share), everyone else gets (larger) PNGs.
That way you don't really care either way on the backend.
Saying 'proper browsers' gives away your bias there, since only the Blink WebKit engine supports it (Chrome/New Opera), which is under 20% of the desktop market worldwide.
Chrome has approximately the same worldwide market share as Firefox on the desktop. Both put together are eclipsed by IE. The metrics that disagree with this stat measure 'hits' (aka pre-2000 web metrics) instead of 'visitors'.
As an exercise, try having your avatar only in the WebP format and see how useful it'll be.