Not if N is bonded to the generator, that is not a floating neutral. That is an important distinction. It does not have to be earthed in order for ground fault protection to work but it will need a bond in the system where N-G are tied together. Running a generator without a N-G bond (floating neutral) does not provide full ground fault protection. G and N do not need to be earthed for ground fault protection. Only bonded at some point.
The reason neutrals are grounded is for ground fault protection. The EGC provides an alternate path back to source to trip a circuit breaker if for example the neutral would become open.
You are correct, it is a single phase of alternating current. 120V is simply the tap in the middle of the transformer turns and neutral provides a path to that point. Very good you understand this.