At least, I’ll try to make it simple.
Air pressure first. So we have three siphons running. One is siphoning water, one new motor oil and the third mercury. All three are in a big vessel. We begin to evacuate the air. Once the pressure in the vessel goes below the vapor pressure of the water, the water boils and that siphon fails. The oil siphon and the mercury siphon continue merrily siphoning. We continue to reduce the pressure, and at some point the oil begins to boil, that siphon fails. The mercury siphon is still working. In fact, a mercury siphon will work just fine in a pretty much total vacuum.
So air pressure is not required. I could go the other way and point out that at some increasing pressure the water would turn solid so the siphon would fail. I could point out that if the ambient temperature were 215 deg. F the siphon would fail at atmospheric pressure because the water would boil.
Think about this one. Take two tubes. For laffs lets make one 6" in diameter. That tube is open on one end, closed on the other. Let’s take another tube, this one 1/4" in diameter. It similarly is closed on one end. Let’s install valves at the top of each tube. We insert the bottoms of the tubes in water and, using the valves, suck out all the air until the tubes are full of water.
Suppose the top of the 6" tube is 3’ above the water level. What is the pressure at that point? It is atmospheric pressure minus 3’ of water head (in round numbers 14.7 PSI - 1.32 PSI, or about 13.38 PSI).
Suppose the top of the 1/4" tube is 4’ above the water level. The pressure at the top of that tube is atmospheric pressure minus 4’ of water head (app. 14.7 PSI - 1.76 PSI, about 12.94 PSI).
Note that the pressure at the top of the 1/4" tube is lower than the pressure at the top of the 6" tube. Note that the weight per foot of water in the 6" tube is 576 times the weight of water per foot in the 1/4" tube. The total weight of the water in the 6" tube is 432 times the total weight of the water in the 1/4" tube. (This, by the way, is why the chain analogy is incorrect).
Suppose that the tubes are in separate containers so we can arrange the two tubes so their tops are at the same level. If we connect the tubes at the top the water will flow from the 6" tube to the 1/4" tube driven by a differential pressure of about 0.44 PSI, that 1’ of differential head.
Suppose we have identically the same setup with a liquid that doesn’t vaporize at vacuum. Note that the same differential pressure would still exist at the top of the tubes, so the siphon would work just fine in a vacuum.