The way I understand it, force gives rise to acceleration. The integral of acceleration is velocity, and the integral of velocity is position. What if forces gave rise to velocity directly? Or what if they gave rise to jerk (the derivative of acceleration)?
In the $F=mv$ world, momentum and force are equivalent. Push an object hard, and it will instantly begin to move. If gravity acts as a force (it could still be an acceleration if we want a slightly different scenario) then everything close to a mass will move towards it, imploding towards higher densities until the pressure balances gravity. There would not be any orbits since velocities would be pointed straight at the main mass. A dropped ball would follow the equation $x'(t)=-g$, that is, $x(t)=-gt + x(0)$: objects fall with constant speed - in fact, when you throw a ball, as soon as it leaves your hand it will change speed to gravity speed.
In the $F=mj$ world, accelerations would be much softer. You would need to press something longer to make it move. A thrown ball would follow the equation $x'''(t)=-g$, that is, $x(t)=-(1/6)g t^3 + a_0 t^2 + v_0t + x(0)$: if released from rest it will take longer for it to drop but the speed will grow even faster. Presumably it would also keep a trace of the acceleration you gave it: throw it hard upward, and it will not just rise but keep on accelerating for a while until gravity won out.
Overall, one can play this game (which is fun), but the real issue if it makes sense mathematically. The structure of classical mechanics is a kind of ladder where the pairs force-acceleration, momentum-velocity and energy-position link up in the right way, producing important things like conservation laws. These other scenarios basically needs to move all physics up or down a rung. That way one could presumably define Lagrangians and Hamiltonians in an equivalent way for the scenarios, getting a very different dynamics and conserved quantities but essentially keeping the structure invariant.