JW, are you agreeing or disagreeing with what Reach4 is saying? I can't tell
If you do away with the neutral then the secondary of the transformer would look just like the primary, one winding. When the positive arc in the sine wave enters the winding the winding becomes a magnet With a north and south pole as the sine wave returns to zero and enters the negative part if the sine wave the north and south of that electro magnet inverts. This happens 60 times a second which causes the winding of the secondary to do the very same thing. The center taps in no way changes this action taking place within the transformer.
Take two volt and half cells and connect them in series and spin it around at 60 revolutions per second letting the ends rub against two slip rings and you will have three volts AC at 60 Hz. Hold then in place and it will be three volts DC. Connect a three volt light to these two batteries connected in series and it will burn. Connect a 240 volt appliance across the secondary of the transformer and it will work.
With this three volt bulb burning a center tap of these two batteries and two 1.5 volt light bulbs can burn with one on one cell and the other on the other cell. All three are burning at one time.
Install a center tap on that secondary winding and two 120 volt appliances can be connect with one on one end and the other on the other end at the very same time the 240 volt appliance is in operation.
In order to have electrons flow there has to be a difference in potential or is you please a positive and a negative. If the two 120 volt sine waves were 180 degrees out of phase with each other this would mean that both ends of the winding would have to be positive at the same time and the center tap negative which would change 60 times a second. Also if they are 180 degrees out of phase the 240 volt appliance would not work.
Connect the scope channel A to one end of the secondary and the ground to the other end of the winding you would see one sine wave of about 340 volts peak or 240 volts RMS. Now move the ground to the center tap and a sine wave of only half of the winding or 120 volts RMS. When channel B is connected to the other end we get to see both ends of the winding at the same time, channel A in positive and channel B is negative which is in sync with each other not 180 degrees out of phase.
Using vector math we add the two together to come up with the single phase winding of the secondary. From the 120 volt reverence at both ends we have 240 volts. 120 from neutral to one end and 120 from neutral to the other end, is vector not degree of sine wave. We also use vector math when adding the two batteries, from the center tap to one end is 1.5 volts and from center to the other end we have 1.5 volts. This is the same as with the AC with the exception of the oscillation which AC has, or in other words while one end is positive the other end is negative.