While you're absolutely correct I'd like to raise a point which DKKHGGGj already did. President is also a influencer of sorts and words coming from his/her mouth carry quite a bit of weight. So, in that regard there's at least some impact on the society and general values of the population drift into whatever direction the president chooses to take. Not a huge amount, but more than nothing. And actual impact depends heavily on who's selected and what kind of support base they have.