Again: that force is proportional to distance.
It may be that the empirical evidence does not suggest any underlying physical mechanism.
As I said before, the underlying mechanism is that the gluons that mediate the interaction are themselves drawn together by exchanging gluons. Here is the empirical evidence that gluons exchange gluons. (Point of ego: I am one of the co-authors of that paper.)
When I went to school the strong force was just described as being "very short range" but I don't think there was much to suggest the mechanism of the force or a mathematical law for it.
You are confusing the inter-quark force with the inter-hadron force. The inter-quark force is proportional to distance, and it is what we mean nowadays when we talk about the strong force. The inter-hadron force is very short-range; it is described by the Yukawa potential. It is what people used to mean when they spoke of the strong interaction, but now we know that it is only a secondary, residual force derived from the inter-quark force (i.e., the true strong force) that holds the individual hadrons together.
The mechanism for this force is the exchange of pi-mesons, or pions. Pions are, loosely speaking, quark-antiquark pairs. (You can think of them as representing the quark-antiquark pairs that result from the breaking of the gluon flux tubes.) The short range of the Yukawa force follows directly from the heavy mass of the pion.