According to moon’s numbers, the #3 hitter comes up 44% of the time with 2 outs and no one on in the first inning. But that follows through in the game.
If every hitter has an equal chance of leading off an inning, then the leadoff spot comes up 1st another 144 times, with the same percentages.
So in the first inning, that means 72 times per a 162 game season, the third hitter comes up with 2 out and none on, and in the following innings does so another 57 times. (I’m using moon’s .333 OBPs, but even if you increase them the numbers don’t vary by much).
So that means 129 times per year, the #3 hitter is up with 2 out and no one on. In a 650 PA season, that represents 20% of his plate appearances. If you up the OBP to .400 for both hitters, the number drops to 18.7%. But it’s in the neighborhood of 1/5 of his PA.
The argument rests on whether any other spot in the top 5 exceeds that number. And they don’t solely because the lineup is set up in the first inning.
In the 80%-ish PA where he does come up with men on, the 4th hitter should also bat unless there is a DP. Which is why the third hitter needs to be capable of avoiding them.
The 3 slot isn’t for bad hitters; he should still be one of the best. But because he is more likely than any other hitter in the top 5 to come up with two out and no one on, his run producing opportunities are limited.
And he should be able to stay out of double plays...