Yes, when you have someone standing over you ready to chop off your head. My first reaction and every time I played was. "Thank the gods that dragon appeared and attacked." That dragon was a friend as far as I was concerned. He actually saved my life. It certainly did not inspire me to go out hunting dragons to kill them. All I wanted to do was get out of there. No hero was I if he wiped out all those Imperials that was fine by me.
Been thinking about this and I think the Dragonborn quest line could have been better done by integrating it into the sandbox play. You escape Helgan, Maybe pick up a shout if you do the Golden Claw quest or go to Whiterun and get the quest for the shard. No dashing off to the West Tower but you are hunted by dragons, low level ones to start with. The dragons instinctively know when someone is the Dragonborn. Now being rescued by a dragon is quite different from being hunted by them. That would get annoying. "Why are they hunting me?" "What did I ever do to them?" You finally manage to kill one or at least help to kill one. You absorb the dragon soul. "Hey what happened?" Someone calls you the Dragonborn. The Greybeards do their calling shout. So you now set out to find out what this is all about.
I just never found the Dragonborn quest line particularly important. The civil ware seemed much more important to me. That made sense had a touch of realism to it.
Well said. If the Dragonborn quest line was intended to be the main quest if failed in my opinion. To me the civil war felt more like a MQ. You had to make a choice there Imperials or Stormcloaks. There was moral ambiguity involved. You met refugees fleeing from the war but I don't recall any saying they were fleeing because a dragon attacked their village.
Some of the dialogue after you killed a dragon and absorbed its soul was hilarious. "Wow I never saw any thing like that! You're the Dragonborn." Then a guard tells you to move along, not cause any trouble or be careful. Even insulting remarks depending on the character you were playing. "Hey guys, I just killed this blood dragon and saved you and your town what about a little respect and appreciation?"
In answer to the OP question there are plenty of dragons in Skyrim they just aren't very important.