I have not graded SAT essays, but have graded state essays required for graduation. The issues are about identical with time and training. The readers selected are usually people who have long experience in grading essays as it takes a long time to develop the skills necessary to be quick, consistent, and accurate. I agree that the first sentence is important as first impressions do count though not to the exclusion of the rest of the essay. Go prepared with what is required in an essay, take a minute to think before writing, take two minutes to create a rough plan for the essay, and then begin writing. The three minutes are important to calm down, and to create the best essay you can in the time allotted.
There is no set time. I also have been trained as a reader. The thing you have to remember is that SAT readers read one question at a time, and read multiple essays on that topic. They are also carefully trained for accuracy.
As a former grader of SAT essays, I agree with your student that a grader reads the essay quickly. An estimate of a minute is probably accurate for the time I typically spent scoring an essay. The graders realize that the essay is the equivalent of a rough draft and therefore take into account that if a student had time to revise, edit, and proofread, the essay would be improved. So the expectations are not unrealistic. As a grader I was an English teacher with about 20 years of experience in evaluating student writing. The scores are based on rubrics unique to each test; an essay is accessed by using this scoring guide. Although a reader may begin a scoring session reading somewhat more slowly, as he or she reads more and more essays, a certain rhythm is established, and the minimum of time spent is probably 1-2 minutes. The essays, after all, are short because of the time limitation.
Don't worry, however, that an essay may not be evaluated fairly. The readers themselves are regularly accessed to be sure that they are following the rubrics so that all essays are graded consistently. The time limit, though, does mean that a student should try to do his or her very best, beginning with the first sentence, which makes a significant impression on virtually every reader I've ever known. A student can, of course, write his or her way into a stronger essay as the writer develops ideas, but that opening is important.