Optimal structure of cover letter for PhD application submitted directly to PI
I would say you are waiting too long to explain why you are interested in the research group. Your motivation for writing and taking up the faculty members' time should come much earlier in the message—I'd recommend no later than the second paragraph.

I'd also suggest that you might also want to consider sending the message to the Oberingenieur or Akademischer Rat of the faculty members' group; he or she may have a better idea of what vacancies are currently available.

Finally, "cold calling" does not get a high response rate; if there isn't an active opening, most potential advisors won't really respond unless they have to.