Implement a brand new search bar

The search bar component is quite old and the code is a bit messy. Additionally, the UI itself hasn't aged well, and we need to rethink how it’s positioned and how it opens.

My idea is to remove the search bar from the top right corner and implement something similar to GitHub's search functionality. For inspiration, see ninja-keys: [ninja-keys](https://github.com/ssleptsov/ninja-keys).