It's hard to understand what you wrote. "A new fuel"? Fuel is a liquid that you put in the fuel tank. Do you mean a fuel FILTER? Or a fuel PUMP? Or a fuel INJECTOR? Or something else?
On your "pumpe-duse" engine, there is no need to "set the timing" if you service any of the above items. Maybe it's something else ... Timing belt, perhaps? Except that even if the timing is wrong, it won't break "rocker arms" - because those are used for operating the injectors, and those don't care if they are mis-timed. The consequence of those being mis-timed is that the engine doesn't start. Except that the only way those can be mis-timed is if the camshaft timing is wrong. But then what breaks isn't the "rocker arms". It can be the "lifters", or the valves themselves.
Now ... On to what I think you really meant.
"They installed a new timing belt, and did not set the timing. As they tried to start the motor, they broke 4 lifters." ... "They said they did a visual inspection of the piston and there is no damage to the lower end (short block)."
Is that what you really meant? If it is NOT what you really meant, then please explain, and disregard what I am about to say. Otherwise, if it is what you really meant ...
Piston and valve collisions should not result in damage to the bottom end of the engine unless the collision is such that the valve itself breaks and smashes around inside the cylinder. I suspect that if their "visual inspection" said there is no damage, this isn't what happened, and thus, there really is no damage to the bottom end.
What can happen is that even if the valve did not break at that time, the valve stem is weakened, and if not replaced, it may fail shortly afterward. Thus, their offer to replace the cylinder head and valve train is thus completely reasonable. Accept their offer, get a warranty on the repair, then keep on driving and stop worrying. The bottom end of the engine is fine.