How can you contextualize what Bob Mayer says in his blog about the "rule of seven" using in the following link the plane crash example?

Maybe the description from that link was marginal. Look also here: https://bobmayer.com/the-rule-of-seven-for-disasters/

I remember hearing this from NASA following both Challenger and Apollo 13. Not necessarily seven, but numerous failures.

The only rule of 7 that I ever heard is the rule that short term memory can only recall 7 plus or minus two items. Link

That number is arbitrary. The Japanese use a similar rule called the “Five Whys?” to solve a problem. These rules do show that when systems are designed properly it take more than one weak point to have a catastropihc failure.
Most systems have redundancies built in. Some are double redundant some are triple.

By the way, to prove how arbitrary it is try this thought experiment. Take any problem or component failure you see. Got it? Now see if you can blame Obama or Trump for it.
Here, I’ll do one as an example:
The component broke.
Why did it break? Because it was stressed beyond its yield point
Why? Because the metal was weak there
Why? Because the manufacturing process did not see a void in the casting
Why? Because the corporation making did not include an X-ray examination process
Why? Because it was trying to be profitable under Obama’s stiff regulations. Thanks Obama!
or
Why? Because it was trying to be profitable under Trump’s lack of regulations. Trump’s an idiot!

The Five whys, or Seven reasons, or whatever manufacturing rule you like, can eventually be used to place the blame on any underling or vendor targeted to take the fall. This can even be done under the guise of deep investigative analysis.
That is why failures need to be investigated by a team of unbiased experts with access to the tools and records needed and the company PR person.

Seriously, try the game. Name a problem: homelessness, hunger, plane crash, etc. Now see if you can use 7 steps to blame it on the person to your right. I’m sure you can do it.

Confirmation Bias.

We are good at finding examples that confirm our working hypotheses and mentally ignoring stuff that disproves them. This is often done in a subtle way. Give me and example and I’m confident that I can find an extra step or two to take it to 9.

Does Bob Mayer not owe that truism to Malcolm Gladwell ?

At any rate, I’ve read and heard Gladwell on the subject, but not Mayer. Gladwell does a good job of explaining the idea, and only somewhat overstates it, IIRC.

It sounds (from your summary) like Mayer may over-emphasize the significance of the number seven, for dramatic effect. I’m not familiar with Mayer, but I’m imagining he’s used to a fictional lens on cause and effect.

I’d say there’s a more intelligent/accurate and less dramatic/misleading explanation for what both of them are talking about, and that NO, there is no one magic number SEVEN involved. But there is a core truth that unusual accidents tend to need to involve multiple mistakes, because people naturally try to avoid disasters, and we tend to notice things we don’t expect to happen, and yes multiple things going wrong at once does therefore tend to be what precipitates major disasters. BECAUSE THAT MAKES SENSE IF YOU UNDERSTAND CAUSE AND EFFECT, and NOT BECAUSE SEVEN IS A MAGIC NUMBER.

Of course, a writer wanting to get readers’ attentions, may be unable to resist the temptation to frame evidence in terms of finding (or creating a story about) a pattern of seven being a magic number, but reality is much more complex, involving vast numbers of circumstances, and analysts or fiction writers need to condense that down to something comprehensible to make sense of situations.

It’s BS. Like when famous people always die in sets of three.

At the office some of the more cynical engineers discovered you could work backwards and no matter the problem, you could always blame it on Tanaka san, an imaginary new or inexperienced employee. Even if the problem was truly caused by Iwata san it became Tanaka san’s fault for taking the closer parking space so Iwata san had to walk more and was tired and less careful. You can think of dozens of ways it is Tanaka san’s fault. He took the elevator so someone had to take the stairs making him late, or he ate the last rice ball so Iwata san was hungry, etc.
It was a fun mental exercise – that we could not discuss in public.

Not me three to seven times.