An intermediate approach I've been taking with my own son is to use calculus and initially not bother with either the manipulation OR the formalism. You have something moving, have the position and want the speed. Or vice-versa. You use it in a bunch of places to build out vocabulary and fluency, and build up both the mechanics and the formalism later.

And we derive a few formulas along the way. integrals of x and derivatives of x2 (x^2, depending on your programming language) pop up all over the place.