Use DocSearch's React component (DocSearch v3)

...instead of our current hack using v2.

[Migration guide](https://docsearch.algolia.com/docs/migrating-from-v2)

We also need to migrate [our legacy configuration](https://github.com/algolia/docsearch-configs/blob/master/configs/api-platform.json) to use the new faceting system.

