Recent Searches is always empty

## Description

We're using the latest `@docsearch/js` alpha and the the Recent Searches never update. I spent a few months in your Discord trying to get answers but still haven't figured out why. No one seems to have a clue, and the site is really quite simple.

## Steps to reproduce

1. Go to
2. Search for anything
3. Search as many more times as you want and you'll never see anything in the Recent Searches modal.

## Expected behavior

Like other sites using the same version (TailwindCSS.com for example), I'd expect to see that list update.

## Environment

- OS: All
- Browser: All
- DocSearch version: 3.0.0-alpha-42
- Project Repo:
- Implementation:

Start with the implementation linked in resources/js/app.js#L16-L33 and reproduce the behavior on statamic.dev with DocSearch 3.0.0-alpha-42. Compare the setup with a working site such as TailwindCSS.com, then verify that repeated searches populate the Recent Searches modal.

Written by the indexing model from the issue text.