Could the Message button also be placed on the user admin page please?
In moderation workflow, checking someone’s profile > sending them a PM occur often for a variety of reasons, and having to hop back out to the public profile is a minor bit of an ongoing annoyance.
I will probably clean this up and make it into a theme component, but if you want to use it right away, add the following code to the </head> tab of the Desktop section of your main theme.