D&D is not a simulationist game, never has been. It is a combination of many playstyles to be attractive to the most people. Shared actions are a gamist conceit, purely there for game balance/action economy reasons, so arguing it doesn't make sense is what doesn't make sense. HP, AC, Hit Dice, halflings with 20 strengths, falling 20 miles from the sky rolling 20d6 and taking only 20 points of damage, making a save vs fireball centered on you in a empty room, tons of things in D&D don't make sense because it is a game.
This is your own view, and not universal truth.
All those other things you mention (HP, AC...) are
models and as such they are intrinsically imperfect, but they
do make sense because they almost always
begin as a representation of something. Only that "sense" is relative, and in fact there are even people who can't stand basic HP and AC. I am sure you have something yourself across editions that you don't find sensible enough for your tastes. Sometimes, a rule can indeed be an afterthough, introduce for balance reasons, and perhaps this may be the case with companions shared actions.
My general idea is that a "rule" or "model" can fail, and not just objectively fail because it doesn't work from a mechanical perspective, but it can also fail from a "sense" point of view. That this failure is (more) subjective, doesn't make it any less legitimate. A RPG is not the same as a card game or board game, and the vast majority of RPG players see it as a combination of narrative + mechanics, and if the mechanical rule jams the narrative of a (fantasy) reality, then it's the rule's fault, not reality's fault. Hence it's legitimate to ignore/modify the RAW, just as it's legitimate to ignore/modify the NAW "narrative as written".
IMO, this rule is asinine. It's an example of over balance and it isn't applied in a consistent manner throughout the system.
Now, you might think that it should take time to direct the familiar to attack each round, but the problem I have with that argument is that Page 190 allows for communication and a single free action (some of which are far more complicated). If those actions don't require you to waste an attack why should your trusted familiar? In addition, the find familiar spell allows you communicate telepathically with your familiar making that argument even less convincing. Even the animate dead spell allows you to use a bonus action or issue general commands. Lastly, an independent mount doesn't need to share its actions with a rider, so why should an independent familiar need to share its attacks with a master?
I don't know the exact wording and presentation of these rules, but I've read other discussions about them here at ENWorld, and I feel like the main reason behind this is possibly that the designers do NOT want a player's turn to last twice as long as everybody else's because of running essentially 2 characters rather than 1.
There might also be a balance purpose, such as considering the companion's attacks or spells as an integral part of the PC, and then thinking in terms "with a companion, this PC is essentially casting 2 spells per round / attacking twice as often this turn, so we'd better prohibit this case".
This is only my best guess. They still wanted "pets" for characters because a lot of gamers love them, but essentially they were very afraid that they can drag the game down and be too beneficial, but indeed from a functional/mechanical point of view this kind of rule makes pets near-worthless in combat.
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I have my own way of dealing with the subject, but this is not something that I expect most people to like: simply put,
to let the DM completely control the companions. That's how I've usually always done (except on PbP games). It doesn't slow the game down more than having one additional monster in the encounter. It won't be perceived as doubling a PC's actions per round. Instead, that companion will be perceived as a
party benefit rather than an individual PC's benefit. Then, you can still reinstate the "requires a PC's action" whenever the player really wants to choose exactly what action the pet is going to take, overtaking the DM's initiative but giving up her own action as a cost.