@Composer: Many pedagogs have the tendency to fixate on some single aspect of a subject and pound on it ad nauseum.  This is not limited to music, by any means, but my point is that every "method" and every teacher will do at least a bit of that. 

I can't really comment on the Suzuki method in specific, since I am not particularly familiar with it.  The closest I came to direct contact with the system was when one of my daughters tried taking violin years ago.  Didn't even make it through an entire school year, and most of what she was telling me about her music classes sounded rather nonsensical to me from my viewpoint.  I mostly avoided the topic, since she was at one of those ages where stuff your parents know and do isn't nearly so cool as the class your friends are in.  When it got to the point where the instrument was collecting dust other than lesson days (assuming she didn't skip the lesson that day), I just asked her if she'd rather have what we spent on the rental as extra spending money.  It was obviously an easy decision at that point.  The "fad" had worn off by then.

So, from what I've seen, it is no particular miracle.  Not any more than any other teaching system for any instrument.  If the student does not have their *own* reasons for wanting to learn an instrument, then the motivation won't be high enough or steady enough, and they won't make the necessary effort.  This is true of any subject or field of endeavour, not just music.

I personally don't agree with looking to the teacher or method or whatever for where to place either the blame or the praise.  If a person wants to learn, they make the effort, and they will use whatever tools they have (which I count methods, teachers or reference books as tools) to get to where they want to be.

So far as double-stops, I suppose that they might be more helpful than some things for working on intonation.  It is at least possible to hear if you are playing the note right, I guess.  But so far as I know, they aren't anything that is strictly for more advanced students?

I think probably everyone has their own ideas about what is "most important' on an instrument or what should be learned first.  I don't have much experience teaching anyone but myself.  But usually, when learning a completely unfamiliar new song/piece, the order I go in is to find the beat and play along with that, then identify the key, then any discernible chord or accompaniment pattern that can be used to play along with simple parts, and then work bits of the melody along with all that.   So I suppose I could say that I find the rhythm and timing the "most important" things to learn first.  But that is just what works for me, and may not work well for everyone else.  I learned that approach from surviving in jam sessions when I barely knew any of the songs, and that doesn't seem to be how most people go about it, from what I can tell. LOL

It'd be pretty silly of me to insist that is how everyone should do it, wouldn't it?   You find what works best for you and you work with it. 

@RosinedUp: I do not doubt that it is possible that a person might start with learning basic theory.  I do find it doubtful that most beginners would choose it themselves, though.  I also have some doubts as to how meaningful theory is without an instrument or voice to apply it to as it is learned. 

I remember back in grade school, they "taught us music".   Assorted little tricks like "Every Good Boy Does Fine" and "FACE" for being able to figure out what the notes were on the treble clef so that we could pass quizzes by being able to spell words like EGG and FEED with "notes" on a piece of paper.  That sort of thing.

It was meaningless, and to this day I do not feel it accomplished anything other than to give teachers one more thing to grade us on.  I didn't actually learn to read standard notation music in any useful sense until I was an adult and intentionally taking a class where it was required.  If you have to figure out that a particular note is G by going through "Every Good Boy Does Fine", then you can't play at any reasonable tempo, and your brain will be too busy with that to be able to spare any attention towards actually making the piece you are playing sound decent.  

Music theory is an analysis of how music works.  It is of little use without music to use it on.  I have known far more players who could play well and knew little theory than theorists who could play well.   Music performance predates theory, it isn't hard to figure out which was the chicken and which was the egg there. 

Most folks will learn how to throw a rock before they learn enough calculus to be able to plot it's trajectory.  But learning to plot trajectories first would not automatically make one better at throwing their first rock, since without the development of the muscles and the experience at using them, the rock isn't going to make it very far.

I can see where it is hypothetically possible to learn theory before learning to sing or play music.  I just can't see where anyone would be likely to want to.  To me, it would be later, when one wants to understand or extend into more advanced possibilities that theory becomes really relevant.

To say it another way.. A child can learn to sing "Twinkle Twinkle" without learning any theory.  Learning theory will not necessarily make learning to sing "Twinkle Twinkle" any easier.  I am not sure as I can just accept as a given that it is better to learn theory first.  It might help things make more sense for some people, but I don't think that benefit is necessarily universal for everyone.

Different folks are different.  What is the best way to do anything?  It depends.  LOL  

@Composer: Tough to find complete agreement perhaps.  But some elements of music are common to all genres.  Playing in tune and on time are necessary in every genre I can think of. 

You are correct in that classical is not one of my genres of interest, though I do have respect for it as a genre, of course.  I have never had any dreams of playing in a symphony.  For that matter, I would consider such an aspiration at this point in life to be about as realistic as if I suddenly wanted to be a "formula one" racecar driver.  For either pursuit, I would lack the highly specialized skill set, the equipment, the resources, and the connections to even make an attempt.  More importantly, I would lack the desire and drive to go after those goals. 

I find it somewhat surprising that you feel there is little interest in classical music here, since I generally think of this site as being more inclined to classical and bluegrass/old-time than most other genres.  At least so far as it is inclined to any genres in particular.  But when people post links to playing they found impressive or that they would one day like to be able to play like, I would say it seems to usually be classical if it isn't bluegrass, sometimes perhaps some Celtic. 

Credit to Pierre/Fiddlerman on the point of thinking to provide discussion areas for at least a reasonable little spectrum of genres, and a bit of instruction on blues/jazz/etc in addition to the basic techniques and terms more likely to be found in classical playing.  It may not be extensive enough to be considered a complete course of study, but that would be a bit much to expect considering he has a busy professional schedule and what there is here is more than one finds on most sites.

But the genres favoured by a given online community will largely fall to what the members happen to have the most interest in.  I think that at least in some ways, this is better than if only classical (or any other particular genre) was all that was permitted here.  While there are other websites that are more inclined towards classical performance, from what I gather, you do not care for those either.