The Intel package driver was already purged from my system. I tried reinstalling it to no use. Purged xwayland, too (which I don't use but nevertheless installed). No help. I've also checked dmesg, messages, Xorg.0.log & syslog and syslog and found nothing out of the ordinary in those logs. It just doesn't work.
I'd be happy to revert back to kdm but it's no longer in stretch or sid. My laptop uses a GM45 chipset, circa 2009. Yes, my laptop's old, but I'm typing this message from Windows 10 insider preview which works just fine. I guess my next step is to boot into my Ubuntu Live CD, which I've been finding I've been using a lot more these days. From there I'll copy the last apt-get changes and see what the change logs are on those.