Considering the new info that Jaehaerys and Alysanne had nine children :O, then their tip to the north may have been more than just a stroll to the wall. One of their daughters marrying a Stark would cement GreatJon's "we married the dragons" and Neds "you had the better claim" quote to Robert. Which gives us some indication that the Starks too, have a tenuous claim. Along with the Arryns.
Rhaenrya had more Targaryen blood than Aegon II through her Arryn mother, (who's Targaryen ancestor was either her mother, through Jaehaerys or great-grandmother through one of Aegon I's three daughters).
Then the Martell's (and through them the Tolands), Velaryons (through Laenor and the Hull bastards), Plumms and finally the Baratheons.
Pretty much everyone in Westeros has Aegon the Conqueror somewhere on their family tree.
I wouldn't say everyone since there are likely hundreds maybe even thousands of smaller great/lesser houses that have no Targ blood due to been to lowly in status or some other reason. But your point still stands most of the great houses more thinly like do have Aegon as an ancestor directly or indirectly somewhere on their family trees