It doesn’t do live crawling, so might not be quite what you want, but I built Stork Search (https://stork-search.net) to solve full-text search for static sites.
Today, you’d run a binary as part of a site’s build or deploy process, feeding in the input files. It generates a search index which you deploy alongside your site. The project’s JS library will load that index and turn it into a client-side interactive search interface.
I’d be curious to see if this sounds interesting or workable for you - you mentioned that you don’t want to host your own index, but does that change if “hosting the index” feels similar to hosting an image, instead of spinning up a server?
I’d be interested in building a paid addition that will crawl your site & host the index - you’re probably the 2nd person I’ve seen with that suggestion. Please let me know if you’d be interested in being a beta user.