I found the answer to my prayers.

The following command helped my to figure out that I was missing libnsspem.so.

curl -v --capath /etc/ssl/certs https://www.worldcommunitygrid.org

Apparently nss from Gentoo doesn't provide that library. Ultimately (since I have two other boxes to cross-reference with) I found out that I need to emerge curl with USE="-nss". I re-emerged and connected just fine.

I was forced to use the nss flag on curl because of a new dependency of thunderbird-bin. I'm currently installing thunderbird so that I can unmerge thunderbird-bin.

If I'd been keeping better notes I would have written a bug report with Gentoo. I don't feel like breaking things to get the failure outputs again.

To make a long story short, in Gentoo, Build curl with USE="-nss" .