Tim is picking on Feynman's comment because even though that acceleration distinction tells you why the twins aged differently in this case, it doesn't pull in the equations that you would need to work through to calculate the aging of each child.
And those equations are about more than just the acceleration.

Feynman said that the reason that one twin ages differently was that one accelerated and the other didn't.
So long as you are comparing something that has accelerated with something that has not accelerated at all, that logic will tell you which one aged more.

Tim is picking on Feynman's comment because even though that acceleration distinction tells you why the twins aged differently in this case, it doesn't pull in the equations that you would need to work through to calculate the aging of each child.
And those equations are about more than just the acceleration.

It's ultimately about click bait.

Try this analogy:

The rate at which an odometer reading increases is not directly affected by the steering, just by speed. But the steering on the way between two points affects the total distance accumulated by the odometer, and knowing the complete steering profile allows you to compute it.

So is "steering" the "reason why" the odometer accumulated more distance on the way between two points?

Correspondence:
odometer distance : proper time interval
steering : proper acceleration
the relationship is also inverted: no proper acceleration -> more proper time

Also, this comment did not originate with Feynman. Five years before Feynman was born, Max von Laue addressed this same topic and highlighted the changing reference frame (ie acceleration) as the key discriminant.
But let's face it, between "Feynman's big mistake" and "von Laue's big mistake" are thousands of views.
As @PeroK said, this ultimately boils down to clickbait.

I think it's worth noting, though, since no doubt at least some readers have read Thomas Kuhn, that there is a transition that fields go through which is not really an extension. In physics, that happened more or less when Newtonian physics came along. (Note that here I'm separating physics from astronomy, although in the modern view the two are very closely related.) Before that, we had what might be called Aristotelian physics. Newtonian physics is not an extension of Aristotelian physics; it's a replacement.

A similar transition occurred in chemistry, from concepts like phlogiston and caloric to the Lavoisier theory of combustion and all that followed from that.

This was the kind of transition Kuhn was preoccupied with and called a "paradigm shift". But then he went (IMO) off the rails by trying to apply it to every transition everywhere in science, which of course doesn't work. Relativity is an extension of Newtonian physics, not a replacement. As Steven Weinberg once remarked, if you have one of those T-shirts with Maxwell's Equations on it (I used to have one but it wore out, I should probably get another one ), you might have to worry about it going out of style (which assumes that it ever was in style to begin with, a dubious assumption, but let that pass...), but not about its becoming false.

However, I feel that
-most of the time- the "physics from YouTube videos" and "physics from pop-science"
reminds me of playing the game of "telephone",
where the words are mostly there (maybe packaged into profound catch-phrases)
but the meanings are often distorted enough to be incomplete, ambiguous, or wrong.

"Physics in 100 years" will reference items in physics textbooks and physics papers,
not items from a YouTube video.

For me pop-science (and maybe YouTube if it were around when I was much younger)
was motivation to me to track down-not bypass-those real physics textbooks.
What is the current situation? And in spite of how successful it has been, what are the problems?
While prose and catchy-phrases may be exciting and sound impressive,
they aren't the appropriate tools to solve physics problems and make any real progress.

. As Steven Weinberg once remarked, if you have one of those T-shirts with Maxwell's Equations on it (I used to have one but it wore out, I should probably get another one ), you might have to worry about it going out of style (which assumes that it ever was in style to begin with, a dubious assumption, but let that pass...), but not about its becoming false.

I'm glad it got reduced to 4. (One thing I never found out was whether anyone had made a T-shirt with the tensor formulation, which would have reduced it to two.) Maxwell's original 20 would never have fit on a T-shirt.