Unlike in war, the marginal curve of adding an extra body goes negative in software. And right quick unless circumstances are favorable.
(If you doubt this, imagine a thousand Arthurs charged with implementing a CRUD app.)
First is that adding people slows things down in the short term.
The second is that due to overhead, adding people won't give linear speed up in completion time.
And third, adding people won't help a project that is linear in nature. The common example is using 9 women to get a baby in 1 month.
My years suggest that he wasn't nearly pessimistic enough.
Now, you might say, "I will just bench those other 27 people" and have them play cards or something. In most organizations, though, you cannot do this. Instead, they must be seen to be useful somehow. And if you have 30 people with their hands in a three-person project, no matter how peripherally, failure is all but guaranteed.
I have seen this many times.
Last time I worked at a big company, there was lots of dead wood, but one guy in particular was just a massive liability. Nice guy, tried hard, but utterly incompetent, and (I think there is a term for this?) he was unaware / ignorant of how incompetent he was. He would often check in code that would break the build (team of 300 engineers, large telecom system), he would write and run scripts that would bring computing clusters to their knees (this was late 90s, there are probably ways to mitigate that now), he would consume lots of high-quality talent's time with basic questions, etc.
I literally asked my manager if we could pay Leo to sit home and play video games. OF course, as you said, everyone's gotta look busy.
Here's the punchline: I learned a new term (to me, at the time... not sure I've heard it again) from a greybeard/wizard there -- this guy was a genius. He had a very appropriate term for Leo: negative producer.
Bingo!
So, adding new bodies does incur cost, but the belief that always adding new bodies always adds more cost than adds impetus to the outcome is not always true.
The classic example is "the hump" which was the cost in fuel terms to ship fuel to China, to be able to fly from China to bomb Japan. Economically ruinous, people forget that missions were nonetheless successfully conducted: it was ruinous but the outcome was achieved within limits.
So the MMM does not say "never add a body" it says "know what your incurring in overhead, choosing to add a body"