Alleyways To Torah

It's because of Hamotzei M'Chavaro Alav H'riyah. The burden of proof is upon the one who seeks to take the object out of the hand of someone else. Since the one who finds the object now has the object, the real owner must provide proof that the object belongs to him.

DLB-
If that's true then there's no point to saying ברי here, as long as he brings סימנים and they match he can even be שמא (whether this can occur in reality can be put to question). In other words, ברי would never work on its own in דיני ממונות and I'm not sure that we find that it doesn't.
In addition, though I'm uncertain and would be glad if someone called me on it, המוציא מחבירו generally takes place in the context of דיני ממונות where I own something and you are contesting that ownership. In the case of a שומר אבידה one would have to see whether he even has בעלות (it's not his) and if he does then one would have to analyze the nature of that level, which might lead to pre-יאוש and post-יאוש distinctions.
I think that in order to conserve what Rav Ally is saying one would have to take a different approach. It could be that ברי ושמא by שומר אבידה isn't really the same ברי ושמא we find elsewhere. Generally (once again I'm not sure) we find ברי ושמא in the context of התחייבות, you say you are sure that I owe you something and I say I am not sure that I owe you and I am not sure that I don't, שמא I owe you. There the question is on the whole עצם transaction, or, even if it's a פרט in the transaction, he's not sure whether part of the claim is legitimate and is מתחייב him in reality. In contradistinction, in our case the שומר אבידה definitely knows that he owes it to someone, but he's simply not sure to whom he owes it, it's more of a ספק in מציאות than it is an actual ספק in the דיני ממונות itself (and טביעת עין of a ת"ח would be saying that we accept his word to be מתיר that ספק). The point is that maybe the whole institution of ברי ושמא and its acceptance in דיני ממונות may not be exportable to all contexts.