It's true that in the RAW, you can't purposefully fail a check including your dex check for initiative. But it lacks verisimilitude if it's not allowed. If someone is asked to bend the bars of a prison window or jump across a chasm, they shouldn't be forced to do their best. If you don't want to try hard to bend those bars or clear that gap, you shouldn't have to. Similarly, if you don't want to act as quickly as you can, you shouldn't have to. That's not the same thing as readying which is using your reflexes to respond to a specific trigger. It's simply allowing others to act before you when you could have gone first.
Except that initiative is one of the few checks that is only a check because that's the game mechanic.
In other words, wanting to delay in a round is fine. Forcing the order is not. By doing this, the player is forcing a situation where no PC or NPC can act in between his actions and his familiar's actions. But that lacks verisimilitude as well. It's mechanics because the initiative system is supposed to be simultaneous in the game, not parsed out and segregated. The only reason turns are segregated is because of the mechanics of the system. PCs are moving simultaneously, attacking simultaneously, etc. It's just handled as segregated. This means that unlike other checks, the player is allowing the PC to influence the mechanics out of character.
It might seem like the same check as trying to bend bars, but it's not. It just happens to be a check. The game designers could have made it not a check at all and we would still have an initiative order. The main reason they made it a check is because they want it modified by Dex.
Let's put this another way. If the character (not the player) wants to delay and suck, how is it that he can guarantee in character that he goes before any other PC or NPC "at that point in time"? Mechanically, there is an "end of round". In character, there isn't. Characters do not know about end of rounds or end of turns. Players do. So characters should not know how to react slowly so that they go at the end of the first round. That is not in character. It's not a decision a character should be allowed to make.
Just because you found this cool mechanical loophole does not mean that a DM should allow it. Roll the dice like everyone else. IMO. And I would say the same to the player of a rogue who wants to always make sure that he goes immediately after the fighter so that he always has a sneak attack target. The character should not be able to guarantee that. Combat should be chaotic, not precisely ordered and determined by one or more players at the table.
Now, I could see a DM making a house rule for someone who wanted to do this type of thing where the PC who wanted to delay a bit rolls a D12 instead of a D20 (or some other houserule). But again, it should still be random. There should still be the chance that other PCs and NPCs go between the wizard's and familiar's inits (or the rogue's and fighter's init's).
But no, a player cannot say that he can just do this because he wants to do this. When a roll is involved, he rolls just like every other player. If he wants to lower his initiative, I might allow him to not add his Dex mod. But, that's as far as I would probably go as a DM. A player who is hell bent on this type of cheese in order to get advantage on nearly every round doesn't find traction at my table. There is already a mechanic for a character controlled lowering of effective initiative. It's called readying. The player does not get to manufacture a new way.
Note that going last in initiative does have its downsides.
It doesn't matter. Just because it has a downside does not mean that a player should be allowed to create his own houserule.