Why isn't a polar protic solvent used in the Finkelstein reaction?
The key point is driving the equilibrium where you want it. You could do it by relative reaction rates. The Finkelstein reaction, however, uses a better method: precipitation of one of the products. Acetone is specifically chosen because sodium chloride, bromide and fluoride are insoluble in it while sodium iodide is soluble. So the equation becomes:

$$\ce{R-X + Na+ ({acet.}) + I- ({acet.}) -&gt; R-I + NaX v}\tag{1}$$

Nothing beats the equilibrium-driving force of precipitation.