Your math is correct. Whether the object will start sliding is independent of its mass. It depends only on the inclination angle and the coefficient of static friction.
Take an object of mass 7kg on a rough inclined plane angle 30 degrees.
We are told the block remains at rest.
After a bit of resolving you conclude that mu>0.57
Now if you replace the block with a 14kg one? what happens?
Intuitively you would say it is heavier and therefore more likely to slide down the plane??
But in the algebra, Friction is dependent on the mass and so is reaction force so they cancel.
So are we saying the mass does not make a difference to whether an object will slide down a plane?
I'm confused!
Ah well, it's probably intuitive that a heavier object gets pulled down by gravity with more force. However... it's probably also intuitive that sliding a heavier object requires more force (picture yourself shoving a small chair over the floor, now picture yourself shoving a big sofa over the floor).
So you could also say that these two intuitions cancel each other out.