Side panel follow-ups

[jtpio]

Opening a new issue to keep track of a few items as a follow-up to #6487

• do a pass on the shell to keep its API surface as small as possible (if possible)
• reorganize some of the sidebar components as suggested by @brichet in Left right panel in Notebook view #6487 (comment):

I was just wondering if the SideBarHandler and the SideBarPalette classes should be move to a sidepanel.ts module, for more readability ? I can handle it if necessary.