Regarding rounds - Yeah, it's still there, but it's far less referenced in general practice, that's all. Take some simple spells, for example: Absorb Elements is a reaction spell - you don't cast it actively, but you use your reaction in response to a trigger to cast it, and this can happen at any point, on anyone's turn. The time for the spell is listed as 'one round', because that's the maximum amount of time it could stick around, but the actual mechanics are that it grants you its effect "until the start of your next turn" - that means that if you step into lava on your turn, and use your reaction in response to that fire damage, then you'll have that resistance all through your turn, all through everyone else's turn, until the start of your next turn. If, on the other hand, the enemy mage that goes just before you flings a fireball your way, and you react with absorb elements, you'll only have that resistance a few moments, because it will end after that mage ends their turn and your next turn starts. Nothing if functionally measured in rounds, in 5e - a number of spells specify one round as their duration, because that's the simplest way of denoting that the effects last until the beginning or end of your next turn (the individual spells clarify the exacts), but beyond that, they measure in minutes, or hours, or more; with one exception that I'm aware of, there aren't really effects that last for a stated number of rounds beyond "1", for the above reason. Short-term spells that lasted a variable number of rounds in older editions (such as spells that lasted a number of rounds equal to you CL or such things), generally now last 1 minute as their standard base; ten rounds is more than any combat encounter generally takes, so it's a simplification that benefits everyone in most cases.
Reactions weren't a thing in earlier editions, and they do a lot of work that you may feel was missing - for example, Counterspell is a normal spell, 3rd level, with a casting time of a Reaction (triggered when you see a creature casting a spell); you don't need to spend your turn preparing to counterspell, or dedicate any extra resources or actions to it, other than spending your reaction for that turn, and the spell slot to cast it. It's, ah... much less of a headache and potential resource waste than spell countering was in earlier editions ^.^
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Just an extra note - you might find some others splitting hairs with you over the sleep example; Sleep doesn't offer a saving throw at all, and works differently to most other spells - a legacy, ironically, from older editions. Sleep affects creatures based on their total hitpoints, and any who succumb stay that way until woken or the spell ends, no save-outs. Hold Person would be a better example to illustrate the problem with Larian's spell timing problem; in proper 5e, landing that first save guarantees that the target will lose at least one turn to the hold effect, no matter when you hit them with it, because the save only occurs at the end of their turn... in Larina-verse, the save-out happens at the start of their turn, so enemies always have to fail your save twice in a row (when you cast it, and at the start of their turn) in order for you to get any benefit out of the spell at all; you're effectively always casting control spells at disadvantage, and at super-disadvantage (which does not exist in 5e), if they would already be resistant to your effect.
