Sidetracking a bit, but isn’t it true that Robert /could/ claim the throne through conquest because the Baratheons technically had royal blood because there were some Targaryens intermingled in there?

Well, no. Anyone can claim the throne by right of conquest – that’s what right of conquest means. Robert was acclaimed king by the lords of Houses Stark, Arryn, and Tully because he had royal blood – as a second cousin to House Targaryen, his candidacy offered some semblance of continuity of succession, which is something that feudal societies care deeply about.