Update to previous post(s): there seems to be some confusion as to how R' Shach and R' Chaim could explain the idea of emunah as belief in that is beyond the rationally justifiable when the Rambam uses the word "leyda", to know. The answer is that the Rambam also uses the term "l'ha'amin", e.g. in Sefer haMitzvos he refers to "ha'amanas Elokus". Question: does the Rambam therefore mean that faith is rooted in "yediya" or "emunah"?
R' Chaim's answer is a classic Brisker "tzvei dinim" sevara. The Rambam meant both. The mitzvah that demands that we acknowledge and accept certain yediyos, factual knowledge about G-d that any rational person can see is true. There is an additional level to the mitzvah that demands that we go beyond that basic mininal standard and believe even that which we cannot justify.
I wanted to add a mareh makom to the sefer Derech Mitzvosecha of the Tzemach Tzedek, p. 44b (mitzvah 25 os b) who spells out exactly these two levels of the mitzvah based on the two formulations of the Rambam, ayen sham.
Also, to add another mareh makom, the idea that emunah is innately programmed into our psyche is not a chassidishe vort of R' Tzadok -- it fits perfectly with the Rambam as well. Look no further than the Malbi"m to Shmos 20:1 who echoes almost word for word R' Tzadok's thesis and bases himself completely on the Rambam.
A tangential point: I took my son to a shiur given in these type topics by a talmid chacham and he was amazed -- he had never heard these ideas approached analytically in the way one picks apart a sugya of gemara. "Why don't we learn this in yeshiva?" he asked. Good question -- I told him to ask his Mashgiach. The truth is that you can go through years of yeshiva and come out ignorant of basic tenets of Judaism, which is pretty scary. I don't have a good answer for how to approach the topics systematically (someone needs to write a good collection of sources for high school age kids to learn), but I think it is important for a ben Torah to go through Rishonim and achronim in these areas just like any other area. If you can appreciate why you need to see a R' Chaim in a sugya in Chezkas haBatim, kal v'chomer you should appreciate why you need to see and understand a R' Chaim in the sugya of emunah.