The Kunena team has announce the arrival of Kunena 6.2.5 [K 6.2.5] which is now available for download as a native Joomla extension for J! 4.3.x/4.4.x/5.0.x. This version addresses most of the issues that were discovered in K 6.1 / K 6.2 and issues discovered during the last development stages of K 6.2
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(sorry for loss of context - this was in response to GoremanX's suggestions.)
I tried this (already was configured that way), and didn't see any difference. I do have the Gravatar integration turned on. If I turn that off, I get a 500 error:
Fatal Error: Cannot pass parameter 2 by reference in components/com_kunena/lib/kunena.file.class.1.6.php on line 60
I see that the problem I get (posted above) when I turn off the Gravatar integration is in the Known Issues for the 2.0 release. I tried the simple fix for that issue (trivial edit to the file mentioned in the error message) and the error went away. Now I can turn off all the integrations in the plug-in manager, particularly the Gravatar integration, and I am able to edit my avatar image.