FLIF is not really a format so much as an experiment. There isn't a spec that I can find, nor is there any sort of formal description of the format. Right now, the only available spec is the source code and that leaves a lot of open-ended questions about what to do about inconsistencies in implementations. I hope it develops into something good, but for now it doesn't seem a viable option.
If we're going to use WebP anyway, we might as well drop PNG too in favor of the lossless mode. I don't know of any downsides to this. But mandatory chroma subsampling means WebP can't completely replace JPEG.