Having used both, the experience of building actual UI with Flutter is a breeze compared to building UI in any game engine. I can imagine that most of the usage of Flutter is leveraging the huge amount of work that was already done to get efficient and capable UIs done with just a stack of widgets.
Godot Engine has pretty good UI building tools.
The places where it poses challenges in my experience are high quality typesetting/rich text, responsive UI that requires a wide range of presentations (i.e. different layout structures on mobile vs desktop), and granular control over rendering details.
But for functionality-oriented UI, it's hard to beat its simplicity and scalability, particularly when you want to develop on one platform (e.g. computer) and have everything "just work" identically on other form factors (web/tablet/etc.).
For example, Godot's editor is bootstrapped/built in Godot, and runs natively on Web, Android, and Quest XR among other platforms.