[ui] Use hierarchical `Route` components instead of manual URL parsing

[gdemonet]

Component: ui

Why this is needed:

See #2809 (comment)

Basically, we are only using Route to choose which "page" to display, while elements within a page could also benefit from such URL matching rules.

What should be done:

Use as many Routes as possible, in all pages!

Test plan:

Everything should work the same as before (with some bugs removed, if any)