I backflush to eliminate air pockets by removing the caliper, pumping the piston out pretty far but not past the seal, then pushing the piston back in with a C clamp.
Never tried a syringe.
But if it can't push fluid back to the reservoir, the caliper is not working correctly.
Note that the relief port is blocked in the first few mm of m/c travel so you get brakeline pressure. Anything like a wrong lever or sticky piston that doesn't retract fully will cause no backflow.
