Searching nightly docs is inconsistently slow

Prior to the rustdoc search overhaul in #144476, it was typically the case that searching the nightly docs or nightly-rustc docs would be slow once per day (presumably when reloading the new docs), and any searches after that point would be basically instant (presumably because the index was fully downloaded and cached).

After the overhaul, it is now the case that most doc searches incur an unpredictable multi-second pause before returning any results, and continue to show a distracting throbber for several seconds after results are shown.

There doesn't appear to be any way to “preload” the full search index, which the previous implementation would do automatically.

This results in a much more frustrating experience, because most individual searches are now slow instead of fast.

Start by comparing searches in the nightly std and nightly-rustc documentation linked in the issue, then read the rustdoc search overhaul from #144476. Reproduce the multi-second pauses and lingering throbber, and compare them with repeated searches before the overhaul; done means ordinary searches return promptly without the distracting post-result loading state.