I imagine things would have gone very differently if the first (relatively high-ranking) representative of the military she'd have had to deal with after being sworn in had gone, '43rd in the line of succession? Secretary of Education? Yeah, right.'

Absolutely. To the detriment of everyone concerned (as without Roslin, Adama would have gone for that Last Stand At Ragnar option), but if it had been someone who would have done what Doral expected Lee to do, try to take over from her (by force if necessary) and refuse to acknowledge her authority, that's where the human race would have ended up. And you're right, it's certainly a huge reason why she trusts Lee more than she trusts his father right from the start.