Things I'd like to see from the solution for this:
Focus is moved to within the block chooser after pressing the button. The search input at the top feels sensible (as was implemented by the original fix for this issue)
Focus gets trapped within the block chooser until the chooser is closed
There is an explicit button to close the block chooser that can be reached via keyboard (tab and screen reader)
The button exposes an open/ closed state
The picker can be closed via the escape key and focus returned to the button