Why some cells don't produce purines?
I am going to throw this idea out there. I was unable to find it stated anywhere so far.

There are several very ancient parasites that make their homes inside erythrocytes and neutrophils. Plasmodium is one, the causative agent of malaria. Trypanosoma is another. These things are ancient degenerate eukaryotes and their lifestyle as intracellular parasites is probably ancient too.  

These things cannot make purines either but they need them and they need a lot of them, because they are reproducing fast. If the host cell does not have the machinery to make purines, that machinery cannot be hijacked by the parasite to really crank out the purines.  The best they can do is hijack the salvage pathways these cells have.  

From https://www.ncbi.nlm.nih.gov/pmc/articles/PMC259377/


  The apparent competition between parasites and host cells for available purines suggests that depletion of extracellular purines should be considered as an approach to treating extracellular trypanosome infections.


From https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4405406/


  Plasmodium species parasites, like many other protozoan parasites, are purine auxotrophs, unable to perform de novo purine biosynthesis. They rely on the host to provide purines


That does not explain brain cells  - unless lack of purine synthesis is a defense of brain cells against Toxoplasma, yet another eukaryotic parasite.

Nothing would please me more than for someone to poke holes in this theory. A prediction from this theory: erythryocytes in all animals should lack purine synthesis - including birds, who suffer from a form of malaria and whose erythrocytes do have nuclei.