Why Vishnu has avatars but Shiva and Brahma do not.
Of the Hindu trinity, Bishnu (Vishnu) did not originally have a body. Perhaps for that reason, he was the most prime God of the Pantheon and Shiva and Brahma were merely later day additions. Plus since he did not have a body he was allowed to have tons of incarnations (avatars) but Shiva and Brahma did not have any incarnations because they had body.

This one can infer from etymological phonetics; eg Sanskrit: Bi = without (as in Birata, Birati = discontinuous, break etc) Sh comes from sharira. Bisu, Bishnu etc. (the article explains why the phonetics alter like this) Bishnu = Without Body. Hence avatars are what are descended !

The article was written in 2011, after which many of my own understandings developed, when I realized the other possible meanings are in compliance with what I have been calling the Sun Theory (of phonetics analysis) where all linguistic objects are merely parts or conjugations of a near meaning of Sun in various hypothetical or real languages as they might have existed or formulated. In that analysis Bishnu (Vishnu) is Bi+Sun. where Bi is as in Big or Bahu or Maha. Sun (Lord) the great.

Plus in that unified understanding bishnu means (basa+agni = Life Fire etc) In that unification there are no different Gods but all philosophical forms of Sun (Vishnu is no different from Shiva, because they are both Life Fire = Sun)

All Gods have been formulated since time immemorial from the original Sun (Lord).

Just in short. It needs more attention from me and you to digest the ideas. Here is the article. (which still needs to re-toned)