Why are you acting like those are contradictory ideas? Having a dragonborn parent does mean you have a dragon in your past heritage. The text says any given sorcerer might be the first of a new bloodline, distant relation is not required.
On the other hand, I wouldn't want to be the first of new line of dragonborn. "Hey mum, thanks, I was going to grow as a full dragon, then you meddled with your eggs and now I am a dragonborn. I am going to brood for a long time! But don't worry, I'll be dead before you notice." Much better to be a dragonborn born out of other dragonborn.
The fun thing will be when we'll see a gnome with a Goliath ancestor. "Yes, I can double my carrying capacity, turn Large for 10 minutes and have advantage on Grapple. I am a gnome, looking exactly like any other gnome, but my great-great-grand-father was a goliath".