I think we're talking about two different uses of the word "intuition".
One is basically to grok. To understand so deeply that you don't have to do the calculations anymore - you can just "feel" the truth, based off of the prior work you did.
The other use of intuition is isn't necessarily connected to prior calculation or heavy effort. Whether it's a rule-of-thumb, a gut feeling, a flash of insight or a lateral leap.
Reasoning from first principles isn't a good counterbalance to the first definition, and that's because the person doing the intuiting has already internalized those first principles.
But it is an effective counterbalance to the latter, for various reasons already discussed.
One is basically to grok. To understand so deeply that you don't have to do the calculations anymore - you can just "feel" the truth, based off of the prior work you did.
The other use of intuition is isn't necessarily connected to prior calculation or heavy effort. Whether it's a rule-of-thumb, a gut feeling, a flash of insight or a lateral leap.
Reasoning from first principles isn't a good counterbalance to the first definition, and that's because the person doing the intuiting has already internalized those first principles.
But it is an effective counterbalance to the latter, for various reasons already discussed.