They don't, at least not on their own. When a generator is in load following mode (think standby generator with load and no other sources) the generator will be run to produce what the load needs, and most loads are lagging, so the output of the generator will be lagging. If the load were leading, the output of the generator would be leading.
In generator reference, supplying VArs is lagging and absorbing VArs is leading, just the opposite of what it is in a load reference. I find it better to think of it as when Watts and VArs are in the same direction it is lagging and when Watts and VArs are in the opposite direction it is leading, then you don't need to worry about generator reference or load reference.
Even when the generator is connected in parallel with other sources, the system generally needs VArs, so the generator is run to produce VArs, in other words a lagging power factor.
But the generator itself has no particular preference.