A project which faced the same problem (unsatisfactory search experience in sphinx) not long ago is the Zephyr RTOS, they recently settled on using the “Programmable Search Engine” offering from Google with a fallback/toggle to the builtin search.
They discarded Algolia in the past, although as far as i can tell only for the reason of not wanting a 3rd party tool, which seems to have changed as no satisfactory solution without relying on a 3rd party had been found.
The PR implementing this with a bunch of discussion attached can be found here: doc: Add Google Programmable Search Engine by kartben · Pull Request #60018 · zephyrproject-rtos/zephyr · GitHub
Key differences seem to be that
- Zephyr has a bunch of autogenerated docs (config options, API, etc) which litter the search results a lot and are usually not the best result, so the need for improved search was probably greater there than for ROS
- Zephyr seems to be well funded enough to pay for the google service, IDK how ROS infrastructure costs are handled right now