Not necessarily. Surge protection limits voltage between any two conductors, and doesn't 'send the surge into the ground.'
Most surge protectors clamp surge voltage from H-N, H-G, and N-G. The hot-to-ground clamping works even with no EGC.
I should have said 'some' or 'many'.
Indeed, the newer, better strips will actually have 4 MOV's. The three you mentioned plus one in series. Older, cheaper ones did rely on a good grounding conductor. Most I see are the older cheaper kind.
I don't agree that the H-G clamping can occur with no grounding conductor. More likely what is happening is a common mode fault being dampened by a normal mode shunt using the neutral as a path to ground which should still suppress at least one surge. MOV's will fuse open, are usually one time use devices, and offer no protection after the initial surge if that surge is of any size at all. The MOV in series will act like a fuse and open the circuit to ensure no further surges will get to the load. No orderly shut down, though.
Some of these strips pose more danger than protection. They like to catch fire. The newer ones are not as prone and I haven't heard of one catching fire recently.
Some of the really good units use reactance to introduce impedance into the line. The coils don't fuse open (generally), offer continuous protection and need no grounding conductor to function.
For some fun, buy a couple used strips from yard sales. Dissect them. Then ask yourself how a circuit board the size of a postage stamp with MOV's you could hide under a dime are going to function under a high voltage surge or a hit by lightning.