Hap seems to be an old word for Hafen (i.e. port).
Now the Habsburg is on a mountain and no ports in sight ;)
Also, according to historians, the name probably stems from "Habesburg" (haben = have in English).
There's also a myth that it stems from Habicht (Eurasian goshawk), but that is likely just that, a myth.
In either case, the castle was never spelled with a p, even in the middle ages.
So there's no way to spin this spelling into something that makes sense. It's just wrong.
Nah I just took en.wiki's word for it which is always dangerous );
If I really have to push the theory, it would be an old high German form of Haupt (referring to an cowshead), since a hill can be head-shaped :)
Otoh I would not go against the word of someone who has probably seen the medieval docs
Edit: de.wiki link you gave mentions the ford
Wahrscheinlich ist der Name der Burg vom althochdeutschen Wort hab oder haw abgeleitet, das «Flussübergang» bedeutet. Damit ist eine Furt bei Altenburg gemeint
https://www.museumaargau.ch/en/habsburg-castle
(Winter hours apply?)
There's a middle German word hap (ford) which could make the NYT correct in 1100 :)
It's on par with spelling Koeln Cologne imho pity those guys