There was one more bit of low-hanging fruit: Consider the distribution of scores for the highest inning (and what the heck, the second-highest too, since if I remove that the distribution for the rest of the game looks very simple). Is this different if it happens in the problematic first inning than if it happens in the others? I guessed that it would be.
And as the plots below show, it is: and so is the second inning's distribution. The first inning is twice as likely as most of the rest to be the highest-scoring inning, but the average score is less.
The average for the highest score if it happens in the first inning is 1.5, while for innings 3-8 it slowly climbs from 2.5 to 2.8. Given that starting pitchers typically get relieved somewhere between 4 and 6, I don't see any smoking gun for pitchers being exhausted, which was the guess of mine that started this study.
The first inning looks like a combination of two distributions: that common to the rest of the innings and something else. If I subtract off the "common distribution": well, see the bottom plot below where I did a rough-and-ready subtraction. The low score end of the leftover distribution looks like a simple Poisonnian random sampling. The high end you shouldn't pay any mind to; the statistics are low and the subtraction was crude.
The second-highest score looks pretty similar, though smaller.
So from last post we see that when we take out the two highest-scoring innings the score distribution per inning seems to suggest that the runs are relatively independent (which is pretty odd. Home runs yes, but small ball no). At any rate, it looks pretty clean.
But we seem to have two failure modes: one of which can happen in any inning and gives the other team an average of at least a couple of runs, and an additional problem in the first-inning, which follows same sort of Poisson distribution as an ordinary inning, with about the same relative probability.
So a team fouls up with some random probability (grounders take bad hops, etc) all through the game (except the last inning), though in the first inning they seem about half again as likely to foul up. On top of this there's some correlated foul-up that happens in just about any inning.
And one more thing: the second-highest scoring inning is more than twice as likely to come immediately after the first-highest than at any other time. So whatever the effect is, it can last more than an inning.
I think I've taken this about as far as I can without delving into pitching changes. It might be amusing to compare different years with different rules/equipment ("Jackrabbit balls"), but I'll leave those as exercises for the readers.