Update
I think we found our tree guy. He's a licensed, bonded, certified arborist, has $3 million in insurance, is experienced, and does a lot of work for local cities (yeah, I know they hire incompetent people, but it makes me feel better anyway, for some reason).
He does NOT recommend topping the maple (that's what taking off the dangerous part amounts to), unlike several of the pros who have come out here. He said we could get away with it, and it would make it safer for now, but it would only lead to more expense and likely danger in the future. Removal is a last resort, but given the shape, location, root and trunk rot this tree has, he believes it's our best option.
If we top it now, we'll no doubt have to have someone work on it every few years, and it could present a major problem when we try to sell the house. So, we figure we might as well take care of it now, and focus on the positives of having more sun for the garden, no leaves to rake, fewer worries in storms and happier neighbors.
I've done a lot of research, received many opinions, and this guy's advice seems to jibe with what my impartial sources are saying. Plus, his rate is reasonable ($2400 for removal), he has what seems like the best plan for taking down the pieces without any collateral damage (lowering them on a line to an empty space in our neighbor's yard), and he can get it done very soon (there's no way we're waiting a month or two to address this).
Thanks to everyone who's offered their best advice; I've learned a ton, and sincerely appreciate it! Hopefully we're making the right choice, and it will go beautifully.