Indeed, you cannot "patch up the weaknesses of the other side’s proposition" of someone supporting a sports team or a religious belief. Because there is no real argument, but rather some kind of emotional commitment.
However, if you engage in the exercise a lot, you'll notice that a huge number of debates are around such notions, where ego, identity, and feelings are more important than ideas.
It's certainly true for lots of political beliefs, but also many technical ones where your social situation depends on the result of the debates more than being wright or wrong (e.g: at work).
Not to mention when you are debating, it happens regularly you win not by convincing the other side, but by winning over the popularity contest with the audience of the exchange. This again, has nothing to do with the logic of the discussion.
So even if the technique works for, say, your discussion with your physicist or philosopher friend (in which you probably didn't need it anyway), there is really little value in it if it can't work in the situations with more at stake.
Or to use a more scientific term:
I call BS.