The other thing that's annoying is that it does not let me categorize my saved sessions. All are in one alphabetical list. That might work when you just have a handful, but my list grew to dozens. Meh.
So I checked again today on the official site for an update:
With such an user interface of the site, it would be too much to expect gimmicks in the program. Yes there was an update. Security fixes.
So I googled for what else there is. And apparently there are plenty of wrappers around PuTTY that bring exactly what I want: Tabs and Session Folders. I don't know how they could hide from me for so long - maybe because neither the official site nor the Wikipedia page mention any.
I've decided to go with this open source and actively maintained one: superputty