In older (pre-4th) editions of D&D, there was a general philosophy that the purpose of the system was to simulate the functioning of a certain fantasy world. Thus magical powers would often be limited to once per day -- with the explanation that this is just how magic works -- but almost all non-magical powers could be used at will. The idea is that, if I could use an "awesome uppercut move" against the last orc, why can't I try the same thing again against this orc? There were also no "once per encounter" powers, even of the magical variety, because "encounter" isn't an in-world concept.
The 4th edition of the game changed this by adding once-per-encounter and once-per-day abilities for almost all classes. This was part of a general move in 4th edition away from simulationism and towards a more gamist view of role-playing games. For example, monsters in 4th edition don't obey the same rules as PC's, and the rules are very much designed for monsters to provide satisfying encounters with a minimum of fuss. This made things easier for the DM in most cases, but also led to the famous problem of, for example, orc swordsmen being unable to use bows or indeed any weapons not listed in their stat block. PC's in 4th edition had once-per-encounter abilities because it's a game, and that's how the game works.
The trouble with abandoning simulationism is that it undermines the reality of the world and the immersion of the players. If you know about once-per-encounter abilities but your character doesn't, it puts a barrier between your perspective and that of your character, since you're thinking about tactical considerations that your character doesn't know about. Instead of playing as your character during a battle, you're playing as yourself with your character as a game piece.
4th edition was criticized a lot for its abandoning of simulationism and embrace of game-oriented rules. Not all players disliked this change, but enough did that it became a controversial aspect of the new edition, and contributed to the split of the D&D player base into 4th edition players and 3.5/Pathfinder players.
Part of the goal of 5th edition is to re-unify the D&D player base by eliminating many of the more controversial aspects of 4th edition. For that reason, 5th edition has returned to a more simulationist vision of D&D, where the goal is to provide coherent rules for a world as opposed to rules for a game. That's why there's no once-per-encounter powers in 5th edition.
An escapist article based on an interview with D&D 5e designer Mike Mearls summed it up this way:
Why could a Rogue only pull off his fancy Daily power
once per day? The only answer was because those were the rules of the
game, not because that was how combat ought or should work in the
fantasy setting...But Mearls doesn't believe that most D&D players
want to play that way.
Regarding your question, there's nothing mechanically broken about adding such powers, and you should feel free to add them in your home game. You will have to adjudicate the question of what exactly constitutes an "encounter", and it might hurt some players' immersion and suspension of disbelief, but it might also help to make the game more fun and tactically interesting.