Being a Large PC is probably not a big deal
Largeness is not, in and of itself, a particular problem. The drawbacks to being Large -- like having a hard time taking cover, squeezing a lot, and being unable to block small creatures in combat -- are probably balanced by the benefits -- like being able to grapple bigger creatures and occupying a wider space, which allows threatening opportunity attacks against a bigger area and physically blocking more squares against Medium to Huge creatures.
In 5e, reach is not an inherent property of size, so there's not a major benefit in being able to threaten non-adjacent spaces for opportunity attacks, as long as you keep the standard 5-foot reach intact. While you might be considering giving the Minotaur a 10-foot reach or an ability like the Bugbear's long limbed (which increases its reach when attacking on its own turn), I'd advise against it. Attacking into a larger area really is a significant boost in melee combat, so be cautious about altering that.
You also might run into some issues if somebody wants to use certain spells or effects that hit targets in a short radius from the character, because a size increase vastly increases the number of squares the effect can reach. For example, the Hunter Ranger's whirlwind attack potentially becomes a lot more dangerous when you can affect an area 4 squares on a side instead of 3, and when its user can share small creatures' spaces. Imagine wading into a horde of goblins and then doing a whirlwind that hits 12 of them at once! (This is somewhat counterbalanced by the risk of exposing himself to up to 12 counterattacks if he can't drop them with that one hit, and the fact that they can intentionally set up the same situation to pincushion him... even if he doesn't have any whirlwind-like abilities.)
It's also worth considering that large creatures are much less impacted by the threat of opportunity attacks. A large creature can reach a much larger area while remaining engaged with one target than a medium creature can. (This is a little hard to explain without using pictures.)
The roleplaying difficulties of being too big for the world around them are probably what the players are looking for in asking to play large characters in the first place, so I doubt those actually count as "drawbacks" in this case.
You will need to alter some abilities
You will want to alter a few racial abilities to take the characters' changed sizes into account:
- The Minotaur's hammering horns should only affect creatures the same size or smaller, since the size comparison is a major limiting factor on who you can push around.
- The Centaur will need to lose the first benefit of equine build so as to not count as a Huge creature when moving stuff around.
There are questions you should have answers for ahead of time
A big question is going to be weapon size. The DMG discusses this a bit in the Creating a Monster section, on page 278:
Big monsters typically wield oversized weapons that deal extra dice of damage on a hit. Double the weapon dice if the creature is Large, triple the weapon dice if it's Huge, and quadruple the weapon dice if it's Gargantuan. For example, a Huge giant wielding an appropriately sized greataxe deals 3d12 slashing damage (plus its Strength bonus), instead of the normal 1d12.
A creature has disadvantage on attack rolls with a weapon that is sized for a larger attacker. You can rule that a weapon sized for an attacker two or more sizes larger is too big for the creature to use at all.
You can easily rule that a centaur is still only able to use weapons sized for a Medium character, since their human half is only human sized, with normal human hands and arms. But a minotaur in the Monster Manual is Large and has a greataxe that deals 2d12 damage, which is going to be completely broken if your PC minotaur decides to pick it up and use it. You need to figure out how you'll deal with that right up front, and let the player know what that ruling is going to be before the game starts. (If the players are asking to be large because they secretly think they're going to have 10 foot reach with a double-damage greataxe, you'll want to disabuse them of that notion as soon as you can. They might drop the entire request!)
You also need to consider how you're going to handle it when one PC decides to ride on another, especially with a centaur in the party. You should talk it over with the player ahead of time to decide how that's going to go -- if centaurs find it demeaning to be ridden, then maybe you can just avoid the issue through roleplay. If not, or if the player decides that "normally no, but it's okay if it's Lance", then you're potentially going to wind up with an intelligent mount that can hit like a PC and lacks the main counterbalance to intelligent mounts, which is the DM having it act against the rider's wishes because it's a person rather than a trained animal. (Yes, PCs can act against each other's wishes, but that's much less likely to happen unless your PCs are particularly fractious.)
You'll want to make it clear to the players right up front that you reserve the right to override game effects that are based on the character's size and may have strange interactions with a naturally Large creature, and that you might need to ask them to switch back to being Medium if it turns out there's some unforeseen effect that makes being Large totally broken.