Contrary to @dsfgsho's comment, I think you have everything to gain and nothing to lose by reaching out for feedback. It's admirable that you wish to improve yourself.

As for your reservations regarding this course of action,

1. Had you been chosen, you would be working under this faculty person. You would be in some working relationship where daily communication would take place, and that person would be approachable to you. No number of sieves or layers between you in the recruitment process will diminish that fact.
2. While true that faculty members are busy people, supervising their group members should be large part of what they do. Obviously, one of the most phases in training a group member is choosing that person, and deserves a fair share of their valuable time. That said, you should allow a reasonable amount of time if you expect a reply. Depending on the time of the academic year, that could be as long as a number of weeks!
3. Faculty members' e-mails are clearly published on the department website. There's good reason for that.

I believe that politely stating your aim and apprehensions, as you've expressed in your OP, are more than enough to warrant a courteous and informative response. I also believe that any person who mistreats prospective students who act as you do is not fulfilling their role as educators.

Contrary to @dsfgsho's comment, I think you have everything to gain and nothing to lose by reaching out for feedback. It's admirable that you wish to improve yourself.

As for your reservations regarding this course of action,

1. Had you been chosen, you would be working under this faculty person. You would be in some working relationship where daily communication would take place, and that person would be approachable to you. No number of sieves or layers between you in the recruitment process will diminish that fact.
2. While true that faculty members are busy people, supervising their group members should be a large part of what they do. Obviously, one of the most influential phases in training a group member is choosing that person, and deserves a fair share of their valuable time. That said, you should allow a reasonable amount of time if you expect a reply. Depending on the time of the academic year, that could be as long as a number of weeks!
3. Faculty members' e-mails are clearly published on the department website. There's good reason for that.

I believe that politely stating your aim and apprehensions, as you've expressed in your OP, are more than enough to warrant a courteous and informative response. I also believe that any person who mistreats prospective students who act as you do is not fulfilling their role as educators.