This approach, which sees the Rambam as having been precise, systematic and rigorous, in his restatement of the halakhah in the Mishneh Torah, has roots way further than Brisk or the other contemporary methodologies. It can be found expressed in diverse rabbinic characters going back for centuries.
R. Shem Tobh ibn Gaon (1283 – c. 1330), in his commentary on the Mishneh Torah Migdal Oz (H. Shehitah 6:8) wrote:
ודברי ר"מ ז"ל מקובלים מפי רבותיו נ"ע מפי הגאונים ז"ל וכפי לשון הגמרא
לא פחות ולא יתר דקדק על לשונו כמו שתדקדק על לשון הגמרא
The words of the Rambam were recieved from his masters, and from the
Geonim, and just as the language of the Talmud is precise and doesn't
include more or less than necessary, so to may be scrutinize his
language just as we scrutinize the language of the Talmud
According to this perspective, not only are the Rambam's words precise, they are deserving of the same degree of scrutiny as the words of Hazal!
Discussing a case where the Rambam appears to have overlooked a rabbinic passage in his formulation of the law, R. Yaaqobh Emden (1697 – 1776) wrote (Lehem Shamayim, Sheqalim 8:8):
אמנם על כרחנו צריך שנפרש דברי הרמב"ם ז"ל בענין שיסכימו להלכה ואל האמת,
כי אי אפשר להמאור הגדול שיטעה בדבר ברור . . . ואף אנו יודעין כמה קושי
יש בהבנה זו. ומה נעשה וצער גדול היה לנו להחליט הטעות ח"ו בדברי הרמב"ם
ז"ל. על כן לא נמנענו מללמוד עליו זכות מה, שכך היא חובתינו וכך יפה
In other words, though we must read the Rambam as being in consonance with the halakhah and the truth, it's not possible that the Rambam - the "Great Illuminator" made a simple mistake and overlooked a known/relevant rabbinic passage. We must develop a limmud zekhuth when facing such perplexities, and to do so is fit and proper.
In response to a scholar that sought to resolve a difficulty in the Rambam by alleging that the Rambam was less than exacting in his formulation of the halakhah, the Hida (1724 - 1806) wrote (Haim Shaal, vol. 1, s. 15):
If such approaches are adopted every insignificant student will be
able to offer them, and what value is there in writing such things?
Furthermore, Maimonides' greatness is renowned. A number of
Maimonides' words are difficult to understand, yet the later
authorities found a proper explanation, for all of his words are
carefully formulated. Therefore, if due to the weakness of our
intellect we cannot resolve his words, assume that the fault lies with
us, and if it is empty, it is because of us. This is something that
doesn't need to be said and is obvious, and so have I received from my
teachers that in our day far be it to offer such explanations... The
great authorities of years past were masters of the Torah and they
could say whatever appealed to them, but not us.
In others words, saying the Rambam was imprecise is a cheap way of getting out of a difficulty and doing so detracts from learning. If we apply ourselves, we will come to a resolution just like previous generations that faced such questions.
If you are interested in exploring this topic further, I highly suggest Marc. B. Shapiro's "Studies in Maimonides and His Interpreters" which spends about 85 pages navigating it.