JPEG-XL is based on the Google Pik proposal among other things and they were active in the spec'ing - there is no conspiracy here.
(Safari learned this lesson earlier when they got stuck being the only browser to support JPEG2000, and Microsoft with JPEG-XR)
I'm not aware that they've shown any interest in JXL, so "little" seems unnecessary.
Google, with AVIF
Google is a major contributor to JXL.
AVIF, as an outgrowth of AV1, has support from every major browser publisher, with the possible exception of Microsoft (though they are a governing member of AOM) as they're the only one which isn't shipping AVIF support right now (and as of October): https://caniuse.com/?search=AVIF
AV1 and AVIF is landing in all browsers. They get the AVIF support for free when they implement the AV1 decoder.
No, it isn't free. You have to implement the support for the AVIF container. This is why libavif exists.