I am admittedly biased in thinking of the Brisker derech as a the key to learning, but have been trying to push myself to appreciate and understand more of R' Shimon Shkop's derech. I haven't gotten down the Sha'arei Yosher enough to predict what he might say on a sugya, but at least I can say with some degree of confidence that the word "sibah" will appear somewhere in the explanation. My chavrusa has suffered though some of the first sha'ar with me, and my son has been a guinea pig for learning through some other scattered pieces. This almost led to the unintended consequence of my son coming to the conclusion that R' Shimon was brilliant and that this Brisker stuff did not have much to offer. Baruch Hashem, a few months more maturity in his learning has led him to see the light. This past Shabbos he told me that he found a R' Baruch Ber that addresses the exact gemara we had previously discussed in a piece in Sha'arei Yosher and (of course!) R' Baruch Ber's approach was far superior. Here is the sugya:
The gemara (Bava Kama 102b) discusses a case where a shliach charged with purchasing goods changes the order and purchases a different item. For example, Reuvain is charged with buying wheat for Shimon, and instead purchases barley. According to one braysa, if the price of barley goes up, Shimon still collects a share of the profit. Why? One explanation offered is that this braysa follows the view of R' Yehudah that shinuy eino koneh, and the shliach's change does not make him the owner of the barley.
The Bnei Ma'arava laughed at this explanation -- true, shinuy aino koneh, but the barley seller thinks he/she is selling barley to Reuvain (the shliach), not Shimon. How does Shimon come to own the barley if there is no da'as makneh to sell barley to him?
The gemara retorts: but even if Reuvain correctly carries out his shlichus and buys wheat for Shimon, there is no da'as makneh to sell wheat to Shimon -- the seller only knows about Reuvain and thinks he/she is selling to him!
R' Abahu replied that the point of the Bnei Ma'arava is valid. If Reuvain correctly carries out his charge, he fulfills the criteria of shlichus. Only if Reuvain changes the terms of his charge and is no longer acting as a shliach does the question of seller's intent (da'as makneh) come into play.
A little elaboration before getting to the heart of the problem: The halacha is that zachin l'adam shelo b'fanav, an person can aquire something on another's behalf without being officially appointed an agent provided there is no downside. In our case, even if Reuvain gets the order wrong, since there is only an upside gain, Reuvain should theoretically be able to act through zechiya as Shimon's agent even unappointed. The barley should belong to Shimon, who would share in the profit. So why do the Bnei Ma'arava assume that a lack of proper da'as makneh, the fact that the seller thought he/she was selling to Reuvain and not Shimon, is a fatal flaw in this theory of zechiya that allows Reuvain to act as Shimon's agent, but if Reuvain correctly fills the order as charged, acting as Shimon's appointed agent through the theory of shlichus, the issue of da'as makneh, who the seller thought he was selling to, is irrelevant ? In both cases Reuvain is acting on behalf of Shimon -- shouldn't the same rules of kinyan apply to both scenarios?
The punchline of the sugya and the distinction being drawn opens the door to explaining other issues, but you need this as a starting point. If you are a Telzer, see Sha'arei Yosher end of 7:7. If you are a Brisker, see Birchas Shmuel in Kiddushin 15:4.