Yes, but a tailwind doesn't "accelerate" (in the common meaning of the word, technically any change in speed is an "acceleration") the bullet. It causes it to lose speed more slowly.
For instance, a bullet fired with a 20mph tailwind will decelerate as if it's muzzle velocity were 20mph slower than it really is.
A bullet with a 20mph headwind will decelerate as if it's velocity were 20mph faster than it really is.
Still happily answering to the call-sign Peetza.
The problem, as you so eloquently put it, is choice.
He is no fool who gives what he can not keep to gain what he can not lose.
-Jim Eliott, paraphrasing Philip Henry.