I don't agree that tapering doesn't affect induced drag when the chord is small. According to linear wing theory, the induced drag coefficient of a configuration depends only on 1) the lift it is generating, 2) the shape of the wing's planform (including any twist), and 3) the aspect ratio. If you were to take any model and precisely scale it up or down, the planform/twist of the wing would remain the same, as would its aspect ratio (by definition). As long as you adjusted the weight (or airspeed) to keep the scale model flying at the same angle of attack, the induced drag coefficient wouldn't change. This means that if tapering at one scale reduced your induced drag coefficient by say 30%, the same tapering would reduce your induced drag coefficient by 30% at any scale.

That said, if you taper a wing with a small chord to reduce the induced drag, you could very easily increase the total drag. This is because profile drag, unlike induced drag does depend on the actual size of the configuration, not just its shape. All else being equal, wing sections with smaller chords have a smaller "Reynolds number" which CAN mean higher profile drag. I think what the engineer was trying to get across was that as wing chords start to get below about 12 inches, tapering may not be worth it. Not because it doesn't offer the same induced drag benefit as it would on a larger wing, but because it can introduce other unwanted characteristics (higher total drag being just one possibility).