I don’t know what you mean by “completing his entire move”. If you mean that he isn’t moving a full 60 feet, I don’t understand. There are always situations in combat where a player decides not to use every last foot of his or her character’s movement. For example:
DM: The orcs look like they’re spoiling for a fight. What do you do?
Player: I close the distance between us and attack.
DM: Okay, they’re about 20 feet away. <Resolves attacks> Two orcs fall dead. You have 10 feet of movement left if you want to use it, but moving away will incur opportunity attacks from the remaining orcs.
Player: Hey! Why are you stopping me from completing my entire move?
DM: ???
You seriously don't see the difference between the two scenarios? In my scenario there's no reason from the PC's perspective to slow down or stop moving. The player is not the one choosing to stop before their full movement is done. You are deciding that they can't suspend their movement mid-air.
In your scenario the PC decides to not continue movement because it would expose them to attacks. It's apples and oranges, a false comparison.
We have turns because we need to split up the activities of the PCs and monsters into bite-size chunks. From the PC's perspective they are moving full speed and leaping over a chasm. Other PCs and monsters may be doing stuff at roughly the same time.
Okay, but that’s just putting a narrative you prefer over a narrative you don’t prefer.
I don’t see a conflict between the rules and the story the way you seem to. To me, it isn’t a zero-sum game. The rules are an input into the resulting narrative. I don’t stop the barbarian from using his remaining 10 or 15 feet of movement. He could make a detour on his way to the chasm or move to a different point along its edge. He could even use his remaining movement to hurl himself into the chasm if he wants to. But if his goal is to leap to the other side, and he doesn’t have enough movement to get there, then that declaration and the resolution of that action are going to have to wait until he does have enough movement.
It’s simpler than that.
First, I should think that your turn is when you get to decide how to use your movement, so of course when your turn ends your move has ended as well. But if you’re saying that I don’t consider movement in the fiction continuous from turn to turn, then no, I absolutely do. From a character’s perspective, the end of its current turn and the beginning of its next turn are pretty much a single instant in time. The barbarian could certainly move the obligatory 10 feet at the end of his turn and make the jump at the beginning of his next turn in one of my games because that’s what he was doing immediately before jumping.
As for my ruling (which I believe is consistent with official rulings on the matter of jumping), it’s that if your declared action for your turn requires movement, you must have enough movement on your turn to complete the declared action. The barbarian doesn’t have enough movement to complete the jump, so the jump can’t be part of his declared action for that turn.
I run games using in-world logic from the PCs perspective. Turns do not exist as far as the PC is concerned. Rules do not exist as far as the PCs are concerned, although they do of course inform the players what activities the PC can take. Hit points don't exist, although people have a general idea of how hurt they are. Sometimes that means I don't strictly follow the letter of the rules. I'm a rebel like that.
If a PC is leaping through the air over a chasm, at some point in time they will be mid-air. It doesn't matter if that point in time corresponds to a turn or not.