Following a bit of intuition, I tried tinkering with my Firefox extensions to see whether I could identify any that might be causing trouble. Rather quickly, I narrowed things down to two - 1Password and/or AdBlocker Ultimate. When both were disabled, the login "stuck" after a Firefox restart. Strangely enough, when the two were both re-enabled, the problem did not reappear. Crazy, but so far (all day today and all evening) the login has persisted as in the past.
I have no idea which extension was at fault, or whether it might have been interplay between the two of them, as I disabled both at the same time. I presume some type of stored data in a preference or something got deleted when the extensions were restarted. Just guessing...
Whether or not this fixes my login issue permanently, only time will tell. I'll try to remember to come back here and post more information if it's needed.
Hopefully, this information might be of assistance for others in the future.