While I agree that it’s somewhat bad that there is no distinction between lossless and lossy jxl in the file extension, I think it’s really not a big deal compared to the present situation with jpg/png.
The reason being that if you download a png file you have no idea if its been converted from jpg, if it’s a screenshot of a jpg, or if it’s been subjected to lossy reencoding by a tool or a website upload process.
The only thing you can really do to try and see if the file you’ve downloaded has suffered encoding loss is to do an image search on it and see if there are any better quality versions out there. You’d do the exact same thing with a jxl file.
Copyrights (rights to media content) do not lapse because of failure to enforce them.
Trademarks (the right to call your product a specific name) can lapse if members of the general public start associating it with a type of product rather than your specific brand. This happened with “zipper”, “jet ski”, and “popsicle”. But you can’t sue Grandma Smith because she mistakingly referred to an Xbox as “a Nintendo”.