I get frustrated if soon into the game I notice a much better site one or two hexes from the original start. If the site is really nice, I may restart and move my settler. I never do that based on horses or iron, however, only after a few turns.
In my last completed game, I had a relatively poor capital site. Two hexes away and I would have had three fishes as well as three deer, plus forests and hills. Any tiles lost would be taken by 2nd/3rd cities. As the fish were a way out, I did not find them until later. All through the game it really annoyed me! I prefer to 'cheat' a little rather than suffer that annoyance, frankly.
I am not sure we do get the best site any more. I think we used to, but no longer. In BNW I have had a few almost useless sites with much better ones nearby.