Correct the artefacts appear in the petfood / relics slot.No that has nothing to do with it. However each item in the magic academy is assigned to a specific 1 of the 5 slots. For example, all buildings offered in the MA share a slot. This means that each time the magic academy craft options rotate, you will only get 1 building to craft. Furthermore, if you are in the process of crafting a building then you do NOT get a new option. To maximize the chances of seeing artifacts, you need to have the slot they would appear in open each time the MA rotates. I believe (90% sure) the slot artifacts appear in is the same slot as relics and pet food. So make sure you are not making any of those when the MA rotates.
Last time I measured it the chance was 2% for the phoenix artefact to appear and 1% for the other.
Since they have added a few more different artefacts to the pool I am not sure the numbers are still the same. but on average you should see 1 phoenix artefact each 2-4 weeks assuming you see all rotations.