Fix Remote Nodes missing custom icons

[CrayolaEater]

BEFORE:

AFTER:

This PR also fixes the type hint for Remote Nodes:

Closes godotengine/godot-proposals#2084
Fixes #95209

[AThousandShips]

No reason to use StringName here when one of the cases is a String and the method takes a String

[Mickeon]

I approve of this but I cannot speak on behalf of the debugger team.

It's unfortunate that the Remote Tree is a completely different thing from the opened Scene display, because code duplication like the one in the PR is required. I considered merging the two someday but it's an very, very daunting task.

[jaydensipe]

This will close godotengine/godot-proposals#2084 and #95209 when merged.

[Repiteo]

Thanks!