I suggest an alternative understanding, one which we can wrap our minds around. In the past I have suggested looking at a disagreement for the sake of heaven as the work of complimentary muscle groups like the biceps and triceps which work against each other but together achieve very fine motor coordination. Today I'd like to provide a very simple logical example of how this works based on a nice self contained piece of Gemara:
[Brachot 13a:] Can we read Shema Yisrael in a language other than Hebrew? There is a disagreement between Rabi and the Hachamim. Rabi says we learn from the word והיו they should be, they should be as they are, unaltered into another language. The Hachamim say we learn from the word שמע understand, we should say Shema Yisrael in any language we understand.
Simple enough, each one has their opinion, each supported from the text. However, the Gemara goes on to analyze some of the results of their opinions: If each were to be examined in a vacuum we would find some very interesting results: According to Rabi, we would be considered to fulfill the mitzwah of learning Torah by reading it in any language whatsoever. Otherwise, why would the Torah need a special case here to emphasize that Shema Yisrael must be read in hebrew? According to the Hachamim, we would only be fulfilling the mitzwah of Torah learning if we learned Torah in Hebrew, otherwise why would the Torah need a special case here to emphasize that Shema Yisrael can be read in any language?
But these opinions don't exist in a vacuum. We can't learn from here that Rabi says learning Torah can be done in any language, and conversely we can't learn that the Hachamim say learning Torah must be done in hebrew. Why not? Because Rabi can say the reason the Torah specifies here that Shema Yisrael must be read in hebrew is because one might have understood like the Hachamim understand and thought Shema Yisrael could be said in any language. And the Hachamim's answer? It isn't such a surprise. They could say that the reason the Torah specifically allows Shema Yisrael to be said in any language is because one might have thought like Rabi that the Torah seems to imply Shema Yisrael must be read in Hebrew.
You see? Neither of these opinions stands on its own. They require the existence of the counter-opinion for their own stance to be self-consistent and logical. The existence of the argument allows both points to have a degree of subtlety and specificity that would be entirely lacking if no argument existed.
Perhaps this is the depth and the strength of a machloket (a rational struggle) for the sake of heaven.