David won't like my answer, and I know we've discussed it before, but I think the real answer is that Eudora's dead, and people using it are just going to have to accept it at some point. Eudora was last actively developed in 2006. Windows 7 was still 3 years from being released at that point. I don't think it's really all that unreasonable for places to say they just aren't supporting it anymore. The problems that it has aren't things that can really be fixed without it being developed further, because technology has changed, and the way eudora expects things to work isn't how things work anymore.
I think for most eudora users, Thunderbird is a fairly reasonable alternative. I know David doesn't like it, but back when I switched away from eudora after I spent some time adjusting, I was able to make it work the way I wanted. (I've since given up and moved completely to gmail myself, simply for phone integration ease.)
I'm pretty sure gmail at least by default blocks eudora. They have something on it that will prevent older email clients from connecting, although you can change that setting and it might then work.