can you not have auth left and lib left viewpoints?

Yes, but actually no. The distinction is fundamentally unstable. If the left is constantly questioning power structures, it will inevitably turn to whatever structure the auth left comes up with.

what's wrong with the definition of left and right on the political compass?

It's specially economic left/right, which is almost always defined by taxation, government spending, and social welfare. While leftists usually say social welfare is a good thing, it's not changing the fundamentals of how capitalism works, which is the current dominant power structure that leftists are against.

do auth left not also take issue with the power capitalist businesses have?

Yes, but they usually put something just as bad in its place. You might have heard people saying that the USSR was "state capitalist rather than communist". This means that the workers and customers had just as little say in how things are run than they would under capitalists, only is was directly with the state rather than individual business owners.