The spell Find Familiar clearly states what kind of creatures can become Familiars. These creatures are "Bat, Cat, Crab, Frog (Toad), Hawk, Lizard, Octopus, Owl, Poisonous Snake, Fish (Quipper), Rat, Raven, Sea Horse, Spider, or Weasel". Every single one of these creatures can be found in the Monster Manual under Miscellaneous Creatures. The quote in the Find Familiar spell about "the familiar has the statistics of the chosen form, though it is a celestial, fey, or fiend (your choice) instead of a beast" is for alignment purposes only, and concerns the casters alignment in these regards. It says nothing about making Celestial, Fey, or Fiend familiars; in fact, it clearly states their statistics are the same as those specifically listed under acceptable creatures listed in the Find Familiar spell as found in the Monsters Manual.
Creatures like the Pseudodragon found in the Monsters Manual clearly state that "some Pseudodragons are willing to serve spellcasters as a familiar." It says "Spellcasters", not anyone who can cast the Find Familiar spell. The fact that Path of the Chain specifically states that "you learn the Find Familiar spell and can cast it as a ritual. The spell doesn't count against your number of spells known. When you cast the spell, you can choose one of the normal forms for your familiar or one of the following Special Forms: Imp, Pseudodragon, Quasit, or Sprite. Additionally, when you take the attack action, you can forgo one of your own attacks to allow your familiar to make one attack of its own." The fact that this special class feature clearly states that a Warlock can gain "one of the following Special Forms," and that "you can forgo one of your own attacks to allow your familiar to make one attack of its own;" while the Find Familiar spell clearly states using specifically listed Miscellaneous Creatures only and that they "can't attack, but it can take other actions as normal" says it all. Obviously Warlocks have special features when it comes to Familiars.
Quoting Variant Familiars from the Monsters Manual doesn't count, that is the Monsters Manual used by a Dungeon Master, not the Players Handbook used by Players. The skills and abilities used in the Monsters Manual clearly differ from those found in the Players Handbook. What would be the point of a Players Handbook if Players could just make characters based off of Monsters with absolutely no changes applied, you know, like the Players Handbook clearly does, same with all other official recourses that make Monsters into playable Races.
The Find Familiar spell clearly states what creatures can be used by the spell, and there are no options to use a higher spell slot to gain different familiars listed in the spell description. The Monsters Manual clearly states that certain spellcasters can use special creatures as Familiars. The Players Handbook clearly states those special spellcasters have The Path of the Chain feature, and are the ones capable of using these Special Forms as Familiars. Having these Special Forms of Familiars is literally a feature of a class; and the Non-Warlock Find Familiar spell clearly defines what creatures can be used by the spell, and the Path of the Chain clearly defines what Special Forms can be used as Familiars by Warlocks. Saying otherwise is ridiculous. This is why D&D clearly has a 13+ age suggestion on their products; because they hope by this age people have the required reading comprehension to understand the rules. The rules on this issue are black and white... literally in black and white print! You can homebrew rules if you like, but the RAW on this issue is clearly stated.
Find Familiar is already arguably the best overall 1st level spell in the game. Name any other 1st level spell that's permanent and can grant darkvision or blindsight, use passive perception to search using superior visual and auditory senses while scouting from the ground, air, or under water up to 100' away, use to gain advantage on ability checks, deliver touch based spells without risking the casters health, and fetch. There is no other 1st level spell as versatile as Find Familiar, and in 5E death of a Familiar is just a minor inconvenience of a ritual spell, time, and components; there aren't even any negative repercussions of your familiar dying like in some earlier editions. What more can you want from a 1st level spell... besides also gaining a 3rd level Class Feature too, just for learning a spell.
