I have been teaching college level math since 1977 and without exception, the students who excel at the conceptual and theoretical aspects of college math are those who have been trained in the basics of algebra and trig. Aside from the absolute necessity of these topics (as David Klein mentioned), the students who have mastered the "rote" stuff have learned how to study properly along the way. Many of the students I have these days have been educated using "fuzzy" methods. Aside from not knowing the fundamentals, they have absolutely no idea how to study - they are clueless to the process of acquiring knowledge.
To believe that the Harvard program will foster genuine conceptual understanding is a huge mistake - unless we are content to have some very superficial type of understanding. For example - students who have some notion that the derivative has something to do with slope but can not compute the slope of a polynomial curve and can not graph a curve without a calculator or computer and who have no understanding of the limitations of machines.
Of course, one can not hope to be a good math major unless they are exposed to theory during their first two years of college math. A number of people have expressed some excitement over the HC text. I suppose change can be initially exciting, but keep in mind that no long term studies have been done with respect to the HC text. Also, any text which claims calculus can be understood, on any level, without facility in algebra and trig. is not telling the truth. Remember we see these students in business calculus and it is very bad - my averages have gone down by 30 points in a decade.
Also, I have always stressed conceptual aspects of calculus. What can be more conceptual then nearness on the real line. The transition from this conceptual idea to inequalities involving absolute value is one thing which makes calculus appealing.
For my money the Harvard book kills concepts. Furthermore, it!! is the easy way out because it doesn't require pre-requisites and has no theory.
If I was willing to buy into this program it would simplify my life.
Please think about this real carefully.
Also folks who benefit from reform in terms of book sales and/or production of articles and degrees are not the most objective judges of this material.