Allow users to message owners of private group to request join

Suppose you have a large number of groups and to save clutter in the groups list you want private groups to be hidden, not publicly visible.

But rather than have owners add people in response to a request received by by some other means you would like to enable users to request access to the group using the same dialogue as they get when they click the request button on an invite group.

There is an admin option for a group to allow users to send membership requests:
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but this only works for a group where the visibility is set to logged on users or everyone, the option is greyed out if the group is not visible.

The Request button brings up a modal dialogue which sends a message to the owners of a group.

Is there either another way to bring up this dialogue from a link that could be posted in a category where you might want to alert members to the existence of the private group and request to join it, or is there a simple message address like @groupname-owners that would enable people to message the owners of a private group with a request to join.