Webpage Login Issues — How to Diagnose
@Tetsujin earns the garland.

The iMac's Safari had uBlock Origin 1.16.0 installed. When deactivated in Safari > Preferences > Extensions, Safari now loads the two problematic webpages correctly.

I'll guess that one of uBlock's data streams (it was subscribed to several, in aid of my trying to block persistent cookie-approval popups) was changed sometime last week, and began hobbling Safari. The cookie-popup blocking was very effective, but the impaired usability turned out to be a cure worse than the disease.