I really don’t see it braking anything as a movement type. As a rogue subclass it would still leave room for a minor benefit imo. It’s on par with spellcasting for arcane trickster, maybe less powerful by a smidge.
Giving a race bonus action teleport of 60 ft has to be on a reset timer, but that’s because it is vastly more powerful than teleporting your normal speed using your movement, bc it’s extra movement that is greater than your normal speed, with a bonus action, on top of bypassing all obstacles.
Take away the added feet of movement all together by restricting it (at least at low levels) to your speed using your movement, and it can be the main part of a rogue or monk lvl3 subclass, or even the lvl1 ability of a warlock patron.
if you’re convinced it’s too powerful, make it less than your speed using your movement. You gain a teleportation speed equal to your walking speed -10ft, or equal to 15ft, or whatever. You can still dash to double that, and monk speed boosts still apply, but it’s much more restricted, and can go up at later levels. On a rogue, it could be feet equal to 5xrogue level, while on a monk it just increases with your normal speed increases.