Some would have you believe that radiation which is not within the main
lobe is miniscual so I suggest
you imagine two models one of which has all lobes removed except the main
lobe Do
you feel that both models now need to be labelled to avoid confusion as
they look so much alike?.

How do you suggest that the power that is not in the main lobe is removed?
You either have to somehow concentrate it into the main lobe or dissipate
as heat in a load.
If you concentrate it into the main lobe then the gain of the main lobe will
go up, if you dissipate it as heat then the gain will remain the same. If
either were possible they would not look like the original yagi.

You seem to miss the point that the gain in the main lobe is effectively the
power in the area illuminated by the main lobe divided by the power in all
other directions.

Reduce the power in the sidelobes then the gain must go up, (unless it is
dissipated as heat, which is not possible in a yagi).

73
Jeff