A potentially major issue I have with AVIF is that because it is based on a video format, any hardware decoding support AVIF will get is likely to be restrained to common video scenarios. This can result in eg. only 4:2:0 YUV being supported by hardware decoders, as that's the upper limit of AV1 Main Profile (and thus the limit of AVIF Baseline Profile). And 4:2:0 is just a poor fit for certain categories of images (like various kinds of illustrations, screenshots, etc). I would not be very happy about a web where lossy 4:4:4 image compression would be heavily discouraged with modern formats.

Also, good lossless compression absolutely does matter for the web as well. Lossy images just cannot be used in all circumstances - like when comparing compression quality (especially of videos), for example!

EDIT: To give an additional example, pixel art is an entire field of art (which is very much still active today) where both lossless compression is a must and where 4:2:0 would be absolutely catastrophic for quality.