They don't seem at odds to me, just two ways of looking at a problem.

They're not so much at odds, but the emphasis is very different and therefore the failure modes are. First principles thinking emphasizes the principles, and so that's why there's an ideological trap; all you need to justify a given direction is some rationale from first principles, and that's often good enough.

The problem is... which first principles? Say your goal is to get to the moon, and the principle you use is to minimize the distance, and you choose to do this in a greedy fashion. This will have you climbing ladders, trees, buildings, and mountains, and you will record that as progress justifying the approach. But you haven't reasoned far back enough to figure out just greedily minimizing your distance to the moon is never going to get you there, no matter what. Choosing the wrong principles can lead to local extremes which can be a lot harder to see in cases less obvious than going to the moon.

Reasoning backwards has its own traps as well.