I would wager it is possible to damage a prop by pushing from the tip. Or at least, I wouldn't take it for granted that it isn't.
Take the example of a helicopter blade... They also develop most of the lift near the tip, yet depending on the rotor type they are designed to allow movement at the blade root, and 'cone' and flap significantly in flight. They are held rigid primarily by centripetal force and if you let the rotor RPM drop too much, the blades may experience structural failure due to excessive flapping.
So the disingeneous answer is that it's fine to push by the blade tips, so long as the propeller is spinning.