Tycho was concerned that the Earth must be too "heavy" and "sluggish" to move. So his system solves this problem. It turns out not to be a problem at all. The problem it solves is "we know that the Earth can't move, so how can we describe the motions of the planets". As the Earth can move, this isn't a problem. So the whole purpose of Tycho's system is gone.
It is a kinematic, not a dynamic model. It describes the motion of the planets but does not give any reason for that motion. This is also the case for Kepler's and Copernicus model. But Newton's model, on the other hand, describes planetary motion in terms of forces.
Now you can do Newtonian mechanics in any coordinate system. It is just more convenient to do it in an inertial system. When you do Newtonian mechanics in a barycentre based inertial frame you derive Kepler's orbits. But if you do Newton's mechanics in an Earth based non-inertial coordinate system, you get Tycho's model. (Or something very like it, but with elliptical orbits)
There are other problems:
- It treats the Earth as "special". This treatment is unjustified. So it isn't isotropic and homgenous: The Earth is still, everything else moves about it.
- It requires the stars to orbit the Earth once a day, this would require them to move faster than light.
- It does not predict things like Foucault pendulum, which demonstrates a rotating Earth.
- It does not predict stellar parallax.
If you start from the axiom "The Earth cannot move" then you must get Tycho's system, or something like it. But if that axiom is rejected then you can get a much simpler model with an inertial coordinate system in which the sun is almost stationary.
Some of these can be "fixed", for example by allowing for a rotating central Earth. But remember the whole point of Tycho's model is to have a stationary Earth. If you can have a rotating Earth, then why not a moving Earth too?