Why the published answer to the "working together" puzzle is wrong

https://mindyourdecisions.com/blog/2017/08/13/can-you-solve-a-math-word-problem-that-stumps-us-college-students-a-working-together-problem/

This thing has gone viral for reasons I cannot comprehend. Here's the scenario:

Alice and Bob can complete a job in 2 hours.

Alice and Charlie can complete the same job in 3 hours.

Bob and Charlie can complete the same job in 4 hours.

How long will the job take if Alice, Bob, and Charlie work together?

Assume each person works at a constant rate, whether working alone or working with others.

The published answer is 1 hour 51 minutes for all three working together.

And clearly, that's wrong. Why the hell would we accept a three-person team that performs only marginally better than a two-person one?

Let's put a little analysis into this thing and see if we can come up with a better answer.

What is the job? For ease of calculation, we'll say the job is assembling 240 objects, and each can be assembled by one person.

When Alice and Bob work together, it takes 120 minutes to assemble them, so the assembly rate in the hands of a competent worker is one minute per item.

It takes 180 minutes to assemble the 240 items if Charlie replaces Bob, and 480 minutes if Charlie replaces Alice. The problem, therefore, is Charlie. Notice it says "each person works at a constant rate" but it does NOT say each person works at the SAME rate. The introduction to the puzzle infers that it takes Charlie three minutes to make one item if he's working with Alice (meaning Alice has to make 180 of them and Charlie only makes 60), and Charlie refuses to work when he's with Bob (meaning Bob has to make all 240 because Charlie spent his whole shift sending tweets even Trump wouldn't chance).

If Charlie won't work at all when Bob is in the room, he's not likely to work if Alice is there too. Therefore, under this scenario it requires all three of them two hours to make the 240 items - 120 for Alice, 120 for Bob and none for Charlie.

Second scenario: Before work starts today, Alice and Bob sit down with Charlie to find out what the deal is. Charlie says he doesn't want to work with Bob and his hands don't work well enough to make the items as quickly as Alice and Bob can. Because the work has to get done, Alice agrees to sit between them and Charlie agrees to pretend Bob isn't there. We know he can make 20 items per hour, so we'll divide the pile of work as so:

Alice: 100 items
Bob: 100 items
Charlie: 40 items

It will take Alice and Bob each 1 hour 40 minutes to do their work. When they are done, Charlie will have finished 33 items. Of the remaining seven, Alice gives three to Bob, takes three for herself and leaves Charlie the last one. Three minutes later, all of them are done. Total time: 1 hour 43 minutes.

Third scenario: Once again, Charlie hates Bob. But Alice watches him build an item and realizes he's doing it all wrong. She gives him a block of instruction that gets his speed up to where it should be. We then divide the pile of work evenly - 80 items per worker. Total in this case: 1 hour 20 minutes.