Well the reason that the neutral wire stays at zero is that it is connected to ground.
The generator (or the local transformer) produces an alternating between two wires. The voltage between them doesn't change if one or the other is connected to earth, so either connection, or even the mid point, can be grounded.
Power is transmitted on both the positive half-cycle and the negative half-cycle, so there is no disadvantage in having one live wire that is sometimes positive and sometimes negative. Having one wire near zero volts means that switches or fuses in the live wire will disconnect the power and make the circuit safe, when there are no voltages relative to ground.
If either wire is not earthed, isolators or switches have to be arranged to disconnect two wires not one, which makes them more complicated.
In situations where positive power only is needed, a bridge rectifier will give voltages that always positive. If a transformer winding is connected to a bridge rectifier, the two ends of a transformer winding will alternately go positive in the way you describe. That is done where DC current is need.