In fact, I'd argue the quaint style of the story does geeks a favor: if it's appealing to normies, maybe they'll appreciate us technical folks' perspective a little more.
lots of people, both technical professionals, and non-engineers who are observant and have an appropriate level of belief in causality, troubleshoot transient failures like this all the time. a difference in the amount of time between shutting down the engine and starting it up is one of the first things that someone like this would test, or control for. it's beyond implausible that the second time the guy got vanilla (after riding along for 4 trips, two long and two short), the engineer didn't raise the question of how long he was in the store.
people troubleshoot things like this by being able to separate causes which are plausible, although unlikely and surprising, from things which aren't remotely plausible. the 500-mile email story and the stories above about sunlight interfering with sensors demonstrate this.
if you're the sort of person who believes that the type of ice cream you get might affecting your car's ignition - the type of person who buys ice cream often but never thinks about how long the errand takes, you simply never get the point of being able to make a pattern between those two things. the second time your car doesn't start, you blame it on the scratch-off lottery ticket you won $2 on which used up your supply of luck for the day. the third time, you conclude that the car ignition knew you were late and likes to choose its failures to cause maximum annoyance. and the fourth time, you realize that your mother-in-law gave your car the evil eye that morning.
the story as told, especially when presented as a real parable about engineering rather than an amusing myth, is frankly insulting to the other type of person. the untrained, not necessarily educated person who cares about machines and believes in material reality. the person who starts checking their watch each time they go to the store and a couple of weeks later is telling their mechanic friend "if it's more than 3 minutes or so, it's fine. but if you try and start it before 2 minutes, then you have to wait another 5 before it's ready to go".
The manufacturer is unlikely to see a problem with putting a display of very popular products right at the front of the store where people can't help but see it.
Or perhaps, unbeknownst to everyone in the world who uses the word 'vanilla' to mean 'mundane', vanilla ice cream actually enjoys far higher profit margins than all other ice cream.
It would be literally impossible for an aisle cap to feature every ice cream flavor available - there are so many that each flavor would have very little representation in the display, and the concept would fall apart as soon as anyone bought something from it. At that point, you're paying a bunch of extra money to send the message "check out our least popular flavors".
That's what I expect them to do, though. The most popular flavors, by definition, needs advertising the least.
The advertising costs the same whether you advertise popular flavors or unpopular ones, but you'll get a lot more sales by advertising the popular ones.
Go check out a grocery store, see whether the aisle caps slant towards popular or unpopular product varieties.
What's not implausible is the idea that one flavor of a product with several flavors might be located far away from all the other flavors. That happens all the time.
And the engineer is sitting in the car on the first night and it "wouldn't start," which signals the end of the episode for the day. Were they stranded? Did the car start after a few tries, which would have given a huge hint about the root cause? Surely the engineer who had reproduced the issue would quickly narrow it down by running diagnoses on the car itself.
But the family dynamics are the most improbable part here. How does the family have this predictable routine and not simply stock up on ice cream? The family has enough kids that the consume a whole $unit of ice cream per day. So with that much chaos in the house, how does the dad justify going out for a drive after dinner when the chaos of family multi-tasking (cleanup, chores, homework, bedtime) is at its peak? "Oh, look. Out of ice cream again. I'll be back in a few!"
Seriously, this is one of those dumbass stories that come from your boomer relatives with the subject line "fwd: fwd: fwd: fwd: fwd: fwd: fwd: fwd: re: fwd: fwd: fwd: fwd: fwd: fwd: vanilla ice cream"
Nobody actually believes this story is true, right?