What is the big problem with a Permanent Pet from a gameplay perspective?

I have always been fascinated by the concept of a champion having a Permanent Pet, and I recently posted a concept for a champion who might have one that can be found here:

https://boards.na.leagueoflegends.com/en/c/skin-champion-concepts/OKMgAfMd-permanent-pet-concept-adaranth-the-narcissictic-mage

But it made me wonder, why has this never been done before? I understand the fact that pets suck to code, but is that the only reason, or is there another reason no one has a permanent pet in game? Is it too hard for players to get used to? Is it too hard to balance?

For the record, I understand that characters like Quinn and Swain technically have pets, but I'm talking about a separate unit, like Tibbers, or Daisy, constantly accompanying a champion.