OMG. It was only after you mentioned the admin link that I was able to find it. Thanks for that.
It’s been 3 days without an answer, so I’ll give you my best guess as someone who has zero control or knowledge, but has been watching stuff here for 6 or more years.
The sidebar is still under fairly active development (It’s still called “experimental” after all!–and they aren’t even taking feature requests), so though you probably can customize it, whatever you do to customize it will very likely be broken very soon, so for now, you should just live with it as it is; if you’re that kind of person, you can upgrade every day or watch the github commits.
I’m unclear how having it be a button rather than a link would change anything? Would the button do something other than open the admin page as the link does now?