Get rid of the Pidgin's "systray" icon

As part of #8309, it would be nice to stop relying on Pidgin’s “systray” icon:

• keep the “systray” icon but let it be in the floating area where these icons land: unless one keeps that area open all the time, they won’t notice that they’ve received messages (unless there’s some additional notification system)
• use some GNOME Shell extension and/or Pidgin plugin to provide proper notifications; ideally these notifications would not fully expire after a timeout, i.e. some indication that one has unread messages should remain visible
  • guifications: notifications expire by default; can they be made persistent? I (intrigeri) live very well with this limitation for my own use case.
  • Pidgin IM integration: it “[d]isplay[s] Pidgin chats as notifications in the Shell message tray”, which may or may not be what we want here. It’s not packaged in Debian, but at least the same author has been maintaining the package for >2.5 years, with the last commit from 4 months ago.
• keep using topIcons for Pidgin, and hope it does not break more stuff (#10576, #11737 (closed), etc.) than it’s worth

Parent Task: #8309

Original created by @intrigeri on 11741 (Redmine)