You mix the relations between the things.
A force produces changes in the linear momentum.
The acceleration $a = F/m$ produces changes in velocity $v = p/m$.
So, your question should be either why don't we take the force proportional to the linear momentum, or why don't we take the acceleration proportional to the velocity.
Now, the second thing is proportionality. Sometimes the force is indeed proportional to the linear momentum (the force of friction), and therefore the acceleration that it imposes is proportional to the velocity, and sometimes the relation between force and linear momentum, (analogously between acceleration and velocity), has more complicated expressions.