Search bar sometimes does not open the search popup

**Your Godot version:**
`4.7.stable`
`Firefox 152.0.6 (64-bit)`

**Issue description:**
When switching to a different page in Godot Docs, sometimes the search popup does not open if the search bar is clicked quickly (before the next page has fully loaded).

Typing in the search bar directly opens the search results page: (pre-4.3 behaviour)
https://docs.godotengine.org/en/4.7/search.html?q=gdscript&check_keywords=yes&area=default

Video:

https://github.com/user-attachments/assets/0fae4843-4cbf-438a-af1a-5f3b4464fb65

**URL to the documentation page:**
https://docs.godotengine.org/en/4.7/index.html
https://docs.godotengine.org/en/4.6/index.html
https://docs.godotengine.org/en/4.5/index.html
https://docs.godotengine.org/en/4.4/index.html
https://docs.godotengine.org/en/4.3/index.html
(the popup did not exist in 4.2, 4.1, 4.0, 3.6, etc.)

Reproduce the issue in Firefox by opening the listed Godot Docs pages, switching pages, and clicking the search bar before the next page finishes loading. Compare quick clicks with direct typing and verify that the search popup opens reliably instead of navigating straight to the search results page.

Written by the indexing model from the issue text.