I believe the /groups page was initially spec’d out for public groups only. It was not a “my groups” page (which would be nice by the way).
I fully support your request since it would make group administration easier for group owners (non-admins). I will leave the bug vs. feature decision for others.
Technically group owners have to be members of the group. I wish that were not true, but I believe that is how it works now. The group slugs are listed on the owner/member’s profile page. Not a convenient way to find a group at all but technically there.
Happy to split my feature request off, but if we are looking at the logic of the page, I would like to add some flexibilty.
My additional request also let’s non-owners find their groups (and the display of those groups can be turned off if needed).