I agree with this, and I think for this reason it is best if a new codec gets wide adoption also outside browsers, instead of having the approach of a "web codec" where it suffices that servers and browsers know about it but the wider ecosystem doesn't support it.
WebP in particular, as the name suggests, was conceived as an "image format for the web" and while I think it's good to have the web in mind when designing an image codec, I think it's a bad idea to limit the scope like that. Design decisions like limiting the maximum dimensions and bit depth at the codec level "because that's all the web needs" plus limited attention/focus to adoption outside browsers does lead to the phenomenon you describe where "it doesn't work for everyone", causing the small gain of improved compression to be dwarfed by the huge inconvenience of breaking workflows.
Any new codec of course has this problem even if they do target wide adoption (like JPEG XL): adoption is never instantaneous. It is a fact that the release cycle of browsers is more suitable for innovation than that of most other software, so it does make sense to start there even when it will still cause things to break in software that doesn't support it yet.
To mitigate that, I think it would help a lot of browsers would have a "Save As..." dialog box on images that gives users the choice to save the actual image in whatever format it is in, or to convert it to PNG or JPEG.