Improve searching in rustdoc and add tests
👋 I have made searching in rustdoc more intuitive, added a couple more tests and made a little shell script to aid testing. Closes#63005.
It took me quite a while to figure out how to run the tests for rustdoc (instead of running tests for other crates with rustdoc); the only pointer I found was [hidden in the rustc book](https://rust-lang.github.io/rustc-guide/rustdoc.html#cheat-sheet). Maybe this could be better documented? I shall be delighted to help if it is desirable.