metabophd

Many programs will agree with your PI. Some will not. It's impossible to say because every adcom is different and its members all hold certain biases. They certainly consider the rigor of schools, but the reality is you're applying to programs that can choose from dozens of students who have top grades from top schools and significant research experience in fancy labs. Harvard, Cold Spring, Sloan, MIT, NYU, Hopkins, and UPenn have plenty star students to choose from.
It's not to say you aren't competitive in general-- looking back at your post I do think you're competitive for a number of the programs you listed, and you might have some luck with the reaches. But I still think that if you don't have much luck it's 1) research experience (get more) and 2) GPA (it's fine but is it better than the students you're competing against?). 
I do think a letter of rec mentioning that you're doing well in classes is a good idea, but I don't think 4 LORs is necessary. As for graduating early, a response to that could be "if your grades were lower due to a heavy course load, why didn't you take lighter course load?". I'm not trying to knock your application-- you'd likely get an interview at my program and overall you're fairly qualified to get in to a solid graduate program. Just saying that should you never hear from the more selective schools on your list, those are probably the factors working against you.