Background Portal Support
There has been issues on the past requesting some form of background checking for emails (such as tray/appindicator) but the solutions back then didn't fit well enough the way GNOME was evolving/has evolved. Since then things have improved, mainly with the introduction of the Background Portal as part of XDG Desktop Portal, there are already applications which have adopted it such as GNOME Geary or Easyeffects, and most prominent desktop environments provide support for it.
This could be a great addition to Evolution, as it would allow not only a way for Evolution to check and notify emails while running on the background in an agnostic way but also would open the possibility for Evolution to be auto-started when the user logs in.