Broken SSL certificates
(With great thanks to @Tetsujin for pointing me in the right direction)

Looking up details of the certificates, in browsers that allow doing that and in Keychain Access, showed that all the breakage was traceable to a "VeriSign Class 3 Public Primary Certification Authority - G5" which was untrusted; I haven't figured out why that was, and it wasn't prominently marked as such in KA.
Anyway, in KA I've set this cert to be "always trusted" on SSL, and everything is now loading well, though the other certs calling upon it are marked as "signed by an untrusted authority".