In your question you say they ask for
3 - 7 years of experience
Then you go on and on about skill, and how much better you are than more experience members of your current peers. 5+ years in the industry would tell you from experience that this is not logically a valid argument. And it would also tell you that is it not a significant one to any hiring manager in the industry as well.
Who told you objectively you are so superior?
With about 2 years in the workforce, I doubt you have a significant exposure to a truly objective cross section of the talent in the industry to make a judgement on who you are comparing yourself to, if your current peers are lack-luster, that still isn't a big pat on the back for yourself.
People skills, not technical skills
The industry is littered with cowboy coders that don't take direction, don't work as a team, don't integrate well into corporate cultures, over estimate their abilities and denigrate their peers; basically these people make terrible employees (really re-evaluate your original question as you posed it yourself with this criteria).
Experienced hiring managers can smell these personalities a mile (1.609344 kilometers) away and will dismiss them as candidates outright regardless of their technical qualifications.
You didn't post the 3 - 4 lines of email, so we don't know what you said, but that could have been a real turn off. If I was applying that far over my head I think I would have put more than 3 - 4 lines into my sell.
The two things are not equal in any manner. Experience implies skill and judgement, skill does not imply experience or judgement.
Judgement is Earned
You can be highly skilled but lack the experience to know when to apply that skill. This is what you are not understanding. Reading books and and "knowing" things is NOT the same things as doing things and knowing what works and what does not work in various given situations and how to apply these things.
Judgement comes not from success, but from failures. Most companies want to hire people that have had their failures paid for by previous companies, that is why they require
N+ years of experience, it implies they made all the basic, entry level mistakes already and someone else had to pay for them.
Judgement only comes with experience.
Edison invented the light bulb through experiencing failure, not through some raw in-experienced skill.
When Thomas Edison was interviewed by a young reporter who boldly asked Mr. Edison if he felt like a failure and if he thought he should just give up by now. Perplexed, Edison replied, "Young man, why would I feel like a failure? And why would I ever give up? I now know definitively over 9,000 ways that an electric light bulb will not work. Success is almost in my grasp." And shortly after that, and over 10,000 attempts, Edison invented the light bulb.
Even the most skilled don't achieve success without a proportional amount of failure. This is how good judgement is earned.
###Talent is useless without Judgement
Talent tells you what to do, Judgement tells you what not to do!
For example, if you were a talented new Jazz musician, you could improvise and know when and what to play. But only experience of playing lots and lots of gigs with lots and lots of different people will give you the judgement of knowing when not to play.
Experienced musicians and programmers can tell exactly how experienced someone else is just by what the person does not play or not code.
If you’re incompetent, you can’t know you’re incompetent. […] the skills you need to produce a right answer are exactly the skills you need to recognize what a right answer is. —David Dunning
If you do not understand the principle behind this answer you might want to read the above a few more times with a critical inner retrospective.
If you had more experience in the workforce, you would not need to ask this question, and would already know that demonstrating your skill is more important than documenting it.
There are countless examples of more skilled, but less experienced people being beaten by less skilled, but more experienced opponents in almost every case imaginable.
Even in the world of Chess, a game where knowledge/skill is highly coveted, experience still wins the day in most cases (human vs. human of course).