I guess I don't mind too much if li chess support Edge or not. I use Chrome mostly anyway. But it would be nice to see how well it works. lol. Not too well. Ha.
Nothing much Lichess can do about this short of a complete overhaul, which is not worth doing just for Edge - as you say, Chrome does the job perfectly well.
Status code 303 is a redirect code in HTTP 1.1 from 1999, rfc warns that clients before this time probably doesn't understand it tools.ietf.org/html/rfc2616
Are you sure you are not behind some old proxy/firewall stuff? Otherwise this sounds like a bug in Edge (or them brushing ancient code under a "new" skin), send them a bug-report or code-patch.