I don't understand why Homer doesn't start the GenSet automatically with the Optimized Mode.
HOMER's dispatch logic is 'optimized' in the sense that it attempts to operate the system so as to minimize cost, but with the limitation that it does not attempt to predict the future. The case you describe is one where some simple prediction could lead to a large improvement in the operation of the system, but since HOMER does not predict the future it operates the system poorly. In most systems the ability to predict the future is less important, although it would often be of some use.
HOMER seems to run the generator at minimal or maximum performance
It does not necessarily run the generator at its minimum or maximum output. Each time step HOMER decides on the correct power output of each generator, and it may choose the minimum output, maximum output, on anywhere in between. Most of the sample files available on the HOMER website demonstrate this.