Great question! Yuda, that reply was insulting, and dangerous…
In Torah she baal peh, machlokes develops as an outgrowth of different understandings. Both sides are branches of the root of Torah reasoning, just one focuses on one aspect, and the other, a different one. For example, in the mishnah you quoted… do we reckon rosh Hashanah l’ilonos based on when the physical manifestations of fruit begins (the syrup filling the tree and the internal budding), and when the majority of the year’s fruit-growing rain has finished falling on the 15th, or do we always go by the first of the month wherein such things happen. Both are logical and revolve around the same idea, with two diverging approaches.
Torah she baal peh was not a static set of kitzur shulchan aruch-esque laws. It was designed to be an ever flowing wellspring of thought and logic that will inevitably produce different results depending on the shoresh neshoma of a given chochom. As long as the chochom has a mesorah on how to go through a sugya, has yiras shomayim, and dedicates his life to the procedure, he is assured siyata dishmaya that his psak will reflect one of the 70 “faces” of Torah. Hashem “agrees” to whatever psak the beis din reaches, as can be seen with the oven of rebbe Eliezer, where a bas kol agreed to him, but the halacha did not follow it. They were listening to Hashem more in ignoring the bas kol than had they listened to it, because Hashem wrote “al pi hatorah asher yorucha” and “laav beshomayim hi”.