Of course points systems can affect how drivers approach a race.
Imagaine at Monaco you have a situation where a title contender is in second place, with a driver ahead that is not in the title race himself, then how valuable that overtake is depends greatly on the points system. If it's 1 point for a win, and zero for everyone else (as was suggested the other year with that medals system), then he's going to want to pass really badly. If it's 20 for first, 19 for second, and 4,3,2,1 for the rest of the top 6, he will naturally not risk it so eagerly.
Current and previous points systems fall between those two extremes, of course, but they can certainly affect things in a variety of circumstances.