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Wow your suggestion really helped: Debug toolbar itself was causing this for some reason. How embarrassing. Thanks!
That is rather surprising, because I don't have the same issue with a pretty large production app. I wonder if there's a deeper underlying issue.
There are a number of toolbar bugs at https://github.com/jazzband/django-debug-toolbar/issues/943 involving keywords "slow" and "hang", not sure which it was.
But the meta-issue is, the profiling tools I used were not super helpful in discovering what the issue was.