Thanks for your reply; I've kept autofills disabled most of this year because of this bug: viewtopic.php?t=315855
which I also just reported tonight.
I tried re-enabling autofill just now but no, that doesn't enable LP to work for me on that page, either - not by clicking my eBay username in the dropdown menu nor by clicking the LP field icons in the name/pass fields (I just tried both; the form fields just stay blank and empty).
As for the other steps and requirements: the add-on's installed, its browser icon is red and I'm logged in, I have no Never URLs, the site's not listed as having Flash or Silverlight by my browser, the login field identifiers are present and my username/pass shows up in the dropdown menu, which I think covers everything the troubleshooting page asks for us to check.
The eBay login page in question always worked for me until it was re-designed a month or so ago so I suspected maybe LP hadn't caught up to whatever changes were made that might not work with it.
But if the page is working for you/others I'm truly stumped, except to wonder if it might be an add-on conflict (Firefox Beta 64.0b5 with some privacy-enhancing add-ons).