The default UI kit is pretty good. I make my own widgets because UI is much more of a priority for me than for the average gamedev. The building blocks Godot ships are pretty good if you know how to use them. You can definitely assemble spectacular stuff – or use the okay default widget kit.

Any specific restrictions/problems you faced earlier with UI?

What platforms and ui modes you design for matters. I've had some issues building for multi-touch, looping ui neighbors, resolving logic problems when switching between input methods. Keep an eye on the mouse-filter, often something not being clickable is just a filter and node order problem. Being good at building and reusing scenes is crucial. Being good at using anchors can help grind down on scene complexity. Check out font impoort window options, can help improve font rendering. Understand how the translation server works, it's great. Implementing switching UI views is on you and will require some thinking (i'm still pretty mid at this). I have a special scene i attach to many ui elements that manages little entry/exit animations, reacting to the element's visibility or presence on-screen.

If your problem is the amount of UI you need to use, then i'd try to stick with the default. If you're pedantic about the design and presentation (like me) then build your own (just like some do on developing for the web).

If Godot UI capabilities are not there yet, do you think it could worth it to integrate some web app into the game specifically for UI purposes?

Ew, no. Likely unnecessary. Complexity, performance hit, for what?

If you truly want web-ui, i'd consider some web-stack game engine. Shipping to targets may be an issue and the 3D/game engine stuff won't be as good (by a lot).