It's been a while since I looked. I could have sworn there were at least a handful, but either A. I might be mis-remembering or B. the landscape might have changed since I checked last.
That said, there seem to be quite few who offer cheap if not actually free access. And by "cheap" I mean, on the order of $10.00 (USD) / month or less.
If you do not care about binary files then use https://www.eternal-september.org, it's free.
Not free, but a few dollars for ARPA membership for life is good.
AFAIK the best options are https://www.eternal-september.org/ and https://usenet.blueworldhosting.com/ (read only though).
Web interfaces:
https://www.rocksolidbbs.com/rocksolid/article-flat.php?id=8...
https://www.novabbs.com/rocksolid/article-flat.php?id=715&gr...
Check the alternative hierarchies and connection options while you're at it.
The difference, I suppose, is that Google Groups are run by autopilot with nobody in the cockpit, Eternal September doesn't ask the average user too many questions (it is as accepting as its name suggests), but those small server admins will ask you questions if you start posting crap, and will delete it, and ban the account.
Sorry, but you make it sound like there are many. That allow posting, and free, I think I only know one.