Yes they can, if your story says so
I'm going to challenge the frame of the question. Should any rules from the PH restrict worldbuilding? Should the rules trump the story?
The point is, you're trying to reduce all the magic in your world to a spell list from the PH. Why? These spell lists are named "Cleric spells", "Wizard spells", therefore imply spells available for a specific class. But NPCs and monsters do not have classes in 5e. Leomund invented his Tiny Hut spell, despite of the fact there are no rules about inventing spells in the PH.
Unless your illithid is a player character, you don't have to restrict its lifestyle with the PH.
You shouldn't restrict its lifestyle with the MM either. Yes, in combat an illithid can kill you fast by sucking your brain out. That doesn't mean it can't eat brains any other way. Say, eat it slowly. Eat a half of a brain. MM Actions describes combat moves or special abilities of a creature. It doesn't describe its "normal" peaceful behavior - a creature can eat, play, feel, sing, laugh — without Actions described in its stats block. Additional source books like VGM or XGtE describe additional things about world, but "additional" is the key word here - these books add to the world, not restrict it. You, the DM, is the ultimate source of information about your world.
Fun > Story > Rules
Consider distinguishing game mechanics from game fiction. The DMG, page 235 says:
The rules serve you, not vice versa.
As a DM, you help guide the narrative and bring the world of the adventure to life. From this perspective, the rules is not your straitjacket, but your tool.
Even PH rules can be changed in favor of story and/or players:
"As DM, I allow changes to characters/classes/etc. whenever it serves the story. The beauty of D&D—it’s infinitely customizable"
(Jeremy Crawford on twitter)
So the real question is — this feeding slaves illithid facility, is it good for your story? Would your world be better (in terms of storytelling) with or without it? Ultimately, will it make the game more fun to play? You're the DM. It's up to you to make a decision.
Lorewise, probably no
If your world is meant to be consistent with the D&D lore, then illithids probably can eat a person only once. While devouring a brain, the illithid consumes not only the brain tissue, but also memories, personality, innermost fears and psychic energy, which is necessary for them:
The brains provide enzymes, hormones,
and psychic energy necessary for their survival.
An illithid experiences euphoria as it devours
the brain of a humanoid, along with its memories,
personality, and innermost fears.
(MM, page 221)
A healing spell can hardly restore all these things, so a regrown brain will be probably not nutritious enough.