For everyone else the role of wizard is held a priest.
The greenskin have their shamans as their wizards
The ogres have their butchers who are wizard-cook-priests and the firebellies who are just priests
Dwarfs don’t have priests and therefore no wizards
Chaos Dwarfs DO have priests and those guys are wizards
In Nehekera lich priests are wizards and as clearly evidenced by their name priests
For Lizardmen their wizards are the Slann (Priestly caste), and skink priests.
Skaven are half and half, in that technically they have other wizards but their main WIZARD guys (Grey seers) are also priests.
Is it because elves were the only ones with a secular magical tradition and the empire got their magic tradition from elves.
I think it’s the last thing you wrote. The Elves just taught whatever they could to humanity (except High Magic which was beyond them).
Also the nature of gods outside of Chaos is really confusing for Warhammer fantasy. like it’s implied they exist and are powerful but where do they normally hang out? Can you draw upon their power?
Some of them at least. Khaine can be drawn upon by witch elves to perform miracles without being wizards (By way of the cauldron of blood) and the sword of khaine is a physical object that you can grab and stab people with.
Lileath routinely hands out super powers
Gork and Mork and the Maw are the source of Greenskin and Ogre magic respectively
Sigmarites do miracles
Sotek does stuff sometimes, which is particularly odd because it’s pretty much stated that the skinks made him up.