I used to use this site until I found https://checkip.amazonaws.com/. Switched because I wasn't sure who was behind icanhazip.com and it's tough to beat AWS. Glad to hear that it will likely be maintained for awhile longer!
I think the point is to prevent middleboxes (eg. caching proxy servers) from interfering with the request. Otherwise I don't really see the issue with the ip address being affected by X-Forwarded-For. You can just... not specify the header.