As airspeed builds, the rudder becomes more effective therefore nosewheel steering is less required/desirable. Your elevator is also becoming more effective, amplifying the forces on the nosewheel. This may result in unnecessary force on the gear and increased scrubbing of the tyres. There will also be a drag penalty, as you mention, though it’s only a small control deflection so I suspect it will also be small.
I’ve not heard the term “downwash” applied to this phenomenon but I understand it to be the horizontal stabiliser entering ground effect. This causes an upward force (or absence of the downward force you’re trying to create) on the tail plane which results in the pitch rate reducing.