Why is the force = mass * acceleration rather than mass * velocity? Isn't the force exerted at a given moment rather than over a period of time, which is expressed through acceleration?

Rather than thinking of as ##\vec F = m \vec a##, it may be more instructive to think of it in this form

##\vec a = \frac{\vec F}{m}##

This separates causes on the right from effects on the left. is directly proportional to the applied force. Acceleration is inversely proportional to the object's mass. Forces cause accelerations, they don't cause velocities or momentum. They cause a change in momentum.

Newton's original expression of the law, in fact states that forces cause change in momentum, rather than an acceleration:

##\vec F = \frac{d \vec p}{dt}##

This statement can be arranged to see that force acting over a time period causes a change in momentum

##\Delta \vec p \propto \vec F \Delta t##

The quantity on the right above, force exerted over a certain duration of time, has a name, as well. It is called impulse (##\vec J##), and as you've gleaned in your question, impulses cause a change in momentum (and therefore in velocity, since momentum ##\vec p = m \vec v##).

##\vec J \equiv \int_{t_0}^{t_f} \vec F dt = \int_{t_0}^{t_f} \frac{d \vec p}{dt} dt = \vecp_{f} - \vec p_0 = \Delta \vec p##

This statement is called the impulse-momentum theorem. The triple equals symbol indicates that this is the definition of impulse.

Force is the instantaneous time rate of change of momentum. Impulse is the accumulated change in momentum over a finite (non-zero) time period.