Players are having double movement.
If monster s don't have it too, hit and run tactics will become deadly and boring.
If your allow tactical move for monsters, the position management will become a mess.
Too much interrupt for my taste.
Obviously any change to a system, such as adding a whole new action type, is going to have an impact on many things. If for no other reason than it challenges the status quo. As I've mentioned above, whatever the players can do, the DM can do as well. The reason I included the movement component is because if you see an ally in need of assistance, it doesn't do a whole lot of good to have an action that could help them if you can't get to them in the first place.
Also, in battle, if you are a more fragile type of character (rogue, wizard, ect), you generally are not going to just stand there while a massive orc runs you down and stomps you. It's not realistic. But by the mechanics, that is exactly what a rogue or wizard must do since they can only act on their turn.
Does this style of combat make it more likely of people chasing someone all over the map? Sure. But Smart players/enemies will change tactics in such circumstances. They may decide to escape the battle and lay a trap that prevents players/enemies from moving around like that.
Also, I kind of look at it like a card game. In my experience, games like Exploding Kittens, Magic the Gathering, Yugio, they all have interesting interrupt mechanics. In my experience, such interrupt mechanics are interesting and add elements to the game that would not be present without them. They make a player wait, paying attention until the moment is right, and give them an opportunity to strike. Sure, D&D has that in the form of reactions, but those often require specific circumstances to use. This gives players a new action type that allows them to follow the battle, staying engaged, and look for the moment they can help out. And unlike a reaction or a more direct attack option, this mechanic does not remove the spotlight from the other player who is currently up in the initiative order.