The big problem is that I can't invite people to join a group or go to an event on Facebook who don't have a Facebook account. Facebook is a little like a bank that won't let you cosign on a loan unless you also open savings, checking, and credit card accounts with them.

A "Facebook account" is just a password attached to a verified email address. Of course you can't attach your identity to a group or an event if you refuse to let Facebook identify you.

If you have a Facebook group where personally identifying participants isn't desirable (i.e. you would want non-FB users participating) than you're probably looking for a wiki or Google Sites instead. Facebook has every right not to be a web publishing tool.