The Baratheon family was around before Aegon the Conqueror. The Baratheons were the Storm Kings. The first Storm King is a legendary figure at the same time as Bran the Builder. Bran assisted the first Storm King in building Storm's End in the face of angry gods.
The last Storm King was defeated by Aegon's ******* brother, Orys. With Aegon's blessing, Orys took the daughter of the last Storm King as his bride. Orys also took the Baratheon name, heraldry, lands and titles... except King. Since then, the Baratheons are the Lords of Storm's End... the primary house of the Stormlords.
There have been seventeen crowned Targaryen monarchs. Viserys and Dany would make eighteen and nineteen named Targaryen rulers of the Targ dynasty. But Dany is only the fourteenth generation from Aegon. Assuming that the Baratheons kept pace with the Targs, then Robert would be either the thirteenth or fourteenth generation from Orys.
The only other infusion of Targ blood into the Baratheon family came from Robert's father's mother, Princess Rhaelle. Her father was Aegon V... Robert's great-grandfather.
Personally, I don't think that Robert would forget date raping Lyanna.
If Robert is Jon's father, then Eddard was not hiding Jon from Robert's wrath, but from the Lannisters'. Tywin and Cersei would never have allowed Jon to live had they known this.
We know of two of Robert's sons, Gendry and Edric. Both are tall for their ages. The only other of his bastards to survive infancy is Mya... and I cannot recall her height. All three of them have black hair. Jon has brown hair. Jon is not tall. That does not prove anything, but it starts a pattern...
Why would Robert fume over Rhaegar raping Lyanna if he'd done the same?
Wouldn't it make more sense for Lyanna to make Eddard promise to hire a drunken septon and make Robert swear that he and Lyanna secretly wed before Lyanna's disappearance? Then Jon would have been legitimate. Robert could still have named Joffrey as heir while having a weapon against Lannister misbehavior.
I prefer my theory of Lyanna being raped by Aerys and then being rescued by Rhaegar. Lyanna was at court. Robert was in The Eyrie. Aerys was a sexual predator. Rhaegar was already married and an honorable man. This would make Jon Dany's elder half brother.
I dunno.
Edit: Of course, I see Aerys' bastards everywhere... Cersei, Jaime, Tyrion, and Jon. If you put those with Aerys' trueborn children, Rhaegar, Viserys, and Daenerys, you have seven. Two years ago I tried to figure out the entire story with a unified theory of everything... Aery's seven children were at the heart of the concept, but I could not make it work.