Is it a disadvantage for a generator to supply both resistive and reactive current to its load?
Yes, there are more losses in the second case.  This is because there is higher current flowing thru the system but with the same amount of real power transferred.

The extra current doesn't cause a higher load on the generator itself.  This is because it pushes the generator forward during part of the cycle and backward during another part of the cycle.  These balance out to no net load over whole cycles.

However, there are inevitable series resistances in the system.  These are in the generator windings, transformers, transmission lines, and other places.  More current thru these resistances means more real loss.

So reactive power flowing thru a system doesn't by itself cause more load.  However, it causes more current, which causes more losses because the system components aren't perfect.