"It's the (action) economy, stupid!"
In analyzing D&D 5e, we often talk about the "action economy" -- in short, the team that gets to take more actions has a huge advantage in the fight.
This is why "boss monsters" are difficult to balance; a big scary monster that gets only one action to the party's four or five is just at a big disadvantage, even if it hits like a freight train. Fifth Edition features purpose-built "legendary" creatures that get to take extra actions outside their own turn specifically to adjust the action economy back towards a more balanced state, making them a more reasonable threat in a one-on-many fight.
Going the other way, if you dig into the encounter difficulty calculations, the DMG has you multiplying the difficulty of a group of enemies based on the sheer number of enemies. For example, if there are 7 or more creatures in the group, you triple the difficulty of the encounter -- the fact that the enemy team just gets to act a lot more often than the heroes makes them vastly more dangerous than their strict CR would suggest. (This is why "crowd control" spells are important; area damage and delaying effects that can break a fight into multiple waves have an outsized effect on large groups of weak enemies.)
While rerolling initiative every round can make "until your next turn" effects unpredictable, the real problem is the way it impacts the action economy. It makes fights virtually impossible to balance.
In the best case, the players may get a lot of turns in a row as they act late in one round and early in the next, allowing them to take out a lot more enemies before the enemies get to hit back and swinging the action economy in the party's favor.
But the nasty one is when it goes the other way; if the enemy just happens to get two turns in a row before the party can respond, they may very well take out one or more player characters without any apparent way to stop them or retreat, which isn't fun, and getting multiple actions in a row can essentially break the tactical movement system by denying the PCs a chance to respond to enemy maneuvers. And of course, once a player character goes down, the action economy turns against the party -- they've lost 20% or more of their damage output, which can turn into a "death spiral" where the disadvantage leads to the party taking more damage than expected, which leads to further knockouts, which leads to more damage, and so on.
Is that going to happen in every fight, or even all that often? No, it won't. The randomness of initiative rolls makes it likely to spread out pretty evenly, in general. But all it takes is one spate of unfortunate initiative rolls to unexpectedly ramp up the difficulty of a fight, or completely ruin a encounter that was meant to be tough, and it can really suck the fun out of a game session. Keep in mind that most fights in D&D only last a handful of rounds, so two initiative rolls that interact badly can mean almost the entire fight is impacted.
Now, there is a time when this happens in D&D as written: when the party is surprised. When one group is surprised, the other side effectively gets two consecutive turns. And what does the DMG have to say about that? Well, on page 85, there's a list of "situational drawbacks" that can raise the encounter's difficulty by a full step (from Hard to Deadly, for example). The very first one on the list? "The whole party is surprised, and the enemy isn't."
To put that another way, depending on how initiative shakes out in the first few rounds, a Hard encounter could end up anywhere from Medium to Deadly, which is a pretty wild change in difficulty!
Worse, a bad action economy is not as obvious as other similar situations. If the players just can't roll above a 5 today, everyone can see what's going on -- nobody can be that upset about what is clearly just the Curse of the Dice Gods. It's frustrating, but nobody's fault. But a funky series of initiative rolls is a lot less visible, which may make the players feel like they're being mistreated. It's not immediately obvious why they're dropping like flies, so it's likely to feel more like the DM has inexplicably hit the party with an incredibly difficult fight, or is somehow cheating, or just "being mean" in some vague way.
It's just not a good change to make in my opinion. The difficulties are not insurmountable, and a mature group of gamers can deal with it (though every gaming group seems to overestimate how mature they actually are), but it seems like adding a significant time cost at the table that comes with a lot of risk and no real reward. What's the actual benefit of randomizing initiative? The game becomes more random, but that's not an inherent benefit. I'd personally much rather get to my turn faster and keep the game rolling along than stop every ten minutes to redo the initiative slots.