The wake vortex is the shear between the air redirected downward by the wing—which balances the momentum given to the aircraft to counter gravity and keep it flying—and the unaffected air outside the span. By not producing lift near the tips you are effectively sacrificing some of your span, which means you need stronger downwash in the middle and that increases the drag, not decreases.

The most efficient configuration (for given span) is when the lift distribution is elliptical. The lift coefficient reduces towards the tips. There are many explanations for that around the site already. Try starting e.g. with For the elliptical wing, which property is actually elliptically distributed? (Peter Kämpf's answers are fairly well linked together so you'll find further references there).

The wake vortex is the shear between the air redirected downward by the wing—which balances the momentum given to the aircraft to counter gravity and keep it flying—and the unaffected air outside the span. By not producing lift near the tips you are effectively sacrificing some of your span, which means you need stronger downwash in the middle and that increases the drag, not decreases.

The most efficient configuration (for given span) is when the lift distribution is elliptical. The lift coefficient reduces towards the tips, but only gradually and never goes down to zero. There are many explanations for that around the site already. Try starting e.g. with For the elliptical wing, which property is actually elliptically distributed? (Peter Kämpf's answers are fairly well linked together so you'll find further references there).