Some very interesting points here. I am not sure if I would agree with some, but I would like to jab at a question. What does age has to do with anything? In my opinion, adults have more logical capacity and want to learn, whilst the child may not have the passion for learning and may not follow the teacher's advice on practicing.
Simply put, all else being equal, age matters a hell of a lot.
I'd say to any adult, especially one that asks if he can play [and definitely if he says his neighbour is crap after 5 years] that the chances are you probably won't get anywhere either. Most kids don't [so even if you want to believe that age has no bearing, chances are you'll still suck] and far, far, far less adults will.
As you age you get older

See kids, parents and grandparents for some of the more obvious differences between them.
You probably wouldn't deny most of those differences. So what's special about this difference? Is it simply easier to pretend it doesn't exist than, say, wrinkles and hair loss?
e.g You could say 'Some kids are overweight but some grandparents do marathons' but that wouldn't negate the fact that, generally speaking, as we get older we lose our health and fitness.
Of course we can find kids that aren't interested and busy adults, but aren't these just stereotypes too?
Forget piano or sums. Just consider the things that most of us take so much for granted that 'everyone can do' and that you probably think as such that they are easy to learn : talking, and so on.
Kids pick them up far, far easier than adults do. Period. How do we know? From kids that didn't learn them as kids.
There should be no "opinion" needed here, either kids learn faster / better or they don't - that's the alarm bell that you should hear when you say that it's your opinion [or mine - or anyone else's] It's either true or not, but not because of our opinons...and certainly not because 'adults are busy, kids aren't'
It does stink I'll admit. Moreso at 39 than at 9 too. But don't take it too far, it's not saying you reach 18 or 28 or 38 and suddenly lose the capacity to learn anything at all.
The science is answering the questions about why that is - that's where you'd need to look.
The only thing that could possibly suggest that children learn faster is that they simply have alot more time on their hands. Adults have work, family, friends, hobbies, etc.
As a child I had work, family, friends and hobbies and school. Didn't most of us?
Where's your adult logic here? Some adults are busy, some aren't. Some kids are busy, some aren't. You can even be busy for a period, then not for a period, as an adult or a child. I'd suggest that there are many things that could show children learn faster than adults.
As an adult, I pretty much have spent the last 5 years or so doing nothing other than trying to play the piano.
Of course, I'm one sample, so it's anecdotal but I think you'll generally find in these threads that appear, that many of those with an opinion about adults learning the piano can be any / all of the following
(a) Not adults
(b) Not learning the piano
(c) Adults that are learning who haven't been told how much their playing sucks and can't tell it for themselves [cf all those pop idol contestants 'Yes simon, people do tell me I can sing' 'Really?' - so they think they're doing ok [and good luck to them]
(d) Adults that can tell, but are happy with their level [good luck to them too]
(e) Adults that learnt as kids [quite often to a fairly high standard] and are going back to it believing that to be 'from scratch'
(f) The tiny, tiny %age of adult learners that can actually play a bit, but think they suck.
From this, as you note, you get myths and assumptions - but I'd say myths that paint a rosy picture for any adults that ask.
By contrast, having tried to learn as an adult, my playing is 100% crap and 100% learnt as an adult.
Contrary to your own stereotype and ingrained assumption - I'm not busy. I wish I was, because although over the past few years ago I've often enjoyed trying to play, I hate the £$%$ing thing with a passion now. I wish I had something better to do, and should that happen I will gleefully smash the £$%£ing thing into a lot of pieces and then set fire to it.
But, to reach this point, I've had the time, lots of time. I've used that time and played for hours and hours a day. Quite literally, sometimes all day.
The end result - I can't play anything. Not a single piece.
That often confuses, but by that I mean play it, correct notes, correct dynamics, correct tempo and so on. Getting a piece to the point where, if I were to meet some grand famous teacher, they might start blathering about interpretation and so on to really begin to play the piece well. I haven't got 3 blind mice to that standard yet.
Of course, I can a few bars of mostly the right notes in roughly the right order of a lot of pieces. So long as I take enough painkillers.
But, the only thing I have to show for years of practise, is a crippling pain in my right arm that pretty much means I've all but given in.
Along the way, I've tried craft of piano playing DVD, Bernhard's [and others] posts, art of piano playing book, 2 local teachers, the usual books and DVDs 'fraser', 'Neuhaus', 'Fink' and so on.
Along the way I've kidded myself a couple of times that I was getting better, that the pain was going, that I was playing notes with my right hand correctly [whatever 'correctly' is] but that doesn't last.
The latest futile attempt is this :-
https://www.thefundamentalaction.com because I can't play pieces anyway, so I don't even bother. I basically just sit now playing C G in both hands as he describes [or perhaps as he describes - I've no more idea if I'm playing the piano correctly than I have of how to perform brain surgery - except for the fact that my playing always sounds crap and it hurts, 2 things that suggest I'm not.]
I usually play that C G or sixths in both hands for four or so hours a day, avoiding the right hand when it's too painful. Hoping that eventually the pain will go and I will be able to start learning to play pieces. As an adult, assuming that I get to the next stage, I'll have spent 5 years to get to the beginning, to the stage where I can sit down and learn to play a piece, comfortably.
Now, of course, I might just suck irrespective of my age. But I doubt that very much.
Besides, even if it is irrespective of my age, it doesn't suggest the mere act of really, really wanting little else other than to play and, the important part of devoting lots and lots of time to it because I'm not busy, is enough.