Avatars are often very personal.
Owned Characters
Many icon posts were commissioned as avatars for the owner of the featured character, but the e621 userbase at large may not want to use such posts for their own avatars because they do not own those characters and have little connection to them.
Humor and Image Formats
Comedy, meme, and especially reaction image posts often make for great avatars because they ooze personality for users to clothe themselves in. Porn
Users will have split opinions on using porn for their avatar, some users preferring to represent themselves with a more chaste avatar than the content they consume, whereas other users want to proudly display their sexuality in their avatars. Even so, porn often does not make for good avatars deserving of the icon tag because most porn has many small details that are lost at thumbnail size and their thumbnails just don't look like much.
Text
Text in icons should either be relatively large compared to the overall size of post or too small to notice at thumbnail size. Text visible from the thumbnail but difficult or impossible to read at thumbnail size makes for a frustrating avatar-viewing experience for other users because they will feel like they are missing the full context of the image, which is very likely true. Animations
Only tag animations with icon if the content of the animation is relatively static and doesn't change much from the frame e621 uses for the thumbnail.
Scenes
Some posts with a detailed background can function as avatars even at thumbnail size because the background is still understandable and the character still communicates something of interest. The koi fish example here resembles classic koi avatars used in forums around 2010.