Firefox not even attempting to open some sites

Many thanks to everyone who replied; much appreciated. I'm afraid that practicality overcame curiosity: thanks to another discussion in these forums I found a way of updating the browsers in both machines while still keeping all the redesign/reconfiguration tweaks that I'd made over the years, something I hadn't thought possible.

So with both browsers now completely up to date the problem has disappeared, along with a few other more minor ones, which was an added bonus. I hated giving up on the quest for an explanation, though.