I recently repaired my Giant MPH lever, replacing the broken push pin. I bled the whole system, which was also needed. Now the lever feels like it has the right amount of resistance when it engages but.. The lever has lots of loose movement before it does anything, and doesn't move back after being squeezed. The new pads just about stop a spinning wheel, but it's nowhere near safe for riding.

What have I missed?

Did you take the piston out too when you changed the push rod? If so, did you put the return spring back in too, it fits on the piston, but may have fallen off and then fallen out of the cylinder without you noticing.

What's the piston? 😕 After Google coming up dry, this is a first hint of a clue as to what might have gone wrong.

Could it have flown out when the brake lever was wrenched forward and pulled out? That's what broke the original washer and circlip holding the push pin.

The piston is the bit the push rod pushes when you pull the brake.
It's not likely you've lost this just changing the rod and copper washer though but it's worth checking