Actually no...we DONT know there were Dragonborn before Alissia, thats the entire point. There were ppl who Shouted before Alissia, but Shouting is not unique to Dragonborn, anyone can Shout with enough determination and training. As far as history is concerned, Dragonborns are never mentioned before Alissia. Even in the tales of Ysgrammor and his Five Hundred Companions in the Merethic Era. Which is very strange considering you have other things like the death of Indoril Nerevar, which was a total fabrication, and EVERYONE believed it and there was STILL evidence to prove the "offical" story wrong. Same with Tiber Septim and his rise to power, there are STILL conflicting reports about him despite their best efforts.
Nords noted that Dragonborn became famous for killing Dragons and devouring their souls. They were respected because they played an important part in destroying Dragons and could use Th'um without need to learn it the hard way.
However, the Dragons were supposed to have been beaten by Alessia's time, with only some remaining. Additionally, none of the Emperors had anything to do with Soul Eating so obviously the Imperial Dynasties are not where Nords got all that.
Imperial Propaganda claims Alessia is the first one... and as far as they care, she was (as far as Imperials are concerned, the Dragonborn are those who can light the Dragonfires and keep oblivion Gates closed)... but Nordic oral traditions tell a different story. They tell about Dragonborn who, before Alessia was even born, killed Dragons and devoured their very souls... but it is questionable if Imperials would think that their and Nordic Dragonborn are the same thing (Akaviri think it is).
That is totaly incorrect. Martin is very distantly related to Tiber, all the Septims are. I think you need to carefully re-read the Dynasty of the Septims. Dragonborn IS hereditary. Otherwise the Blades would not have obsessed over making sure the Septim Bloodline continued at any cost, since we know their entire purpose was to ensure Dragonborns continued so they would exsist when the World Eater returned.
Only second Emperor was Tiber's descendant actually. The rest are from branch family.
However, I agree that gift can be hereditary. It is just that not everyone awakens the unique Dragonborn abilities I think.