Is there any chance at getting a function explorer window like this?

Absolutely. I've got a plan to enable some helpful UI panels for the PowerShell extension within the next few months. I'll make sure this is a part of it.
Please make it also to support PS classes, memebrs and methods.
Have you tried workbench.action.gotoSymbol (default key mapping Ctrl+Shift+O), then type :?
This will be possible in the very near future thanks to VS Code's new explorer tree API in 1.13.0:
Closing as resolved.
Most helpful comment
This will be possible in the very near future thanks to VS Code's new explorer tree API in 1.13.0: