1) The Gemara tries to prove rov from sa'ir hamishtale'ach. I never understood this. In the other sources of rov, such as eglah arufa etc., we do the action and we have to fulfill the obligation, yet we can't check to ensure it's not a treifa, so it must be we follow rov. But regarding the sa'ir, we don't do anyhing, Hash-m does. THe goral will only choose for lazazel a non-treifa. How does this prove rov? Maybe really rov are treifos, but Hash-m will ensure that the goral picks a non-treifa. In other words, when the matter is done by Hash-m, you can't prove anything.
Also, what si pshat when the Gemara tries to defelct the proof and suggests that we check the sa'ir hamishtaleach to see if it was a treifa. We just said the goral will only choose a non-treifa. This is a metzius. How is it shayach to say, maybe there really is no rov, and we check, we just said it is certainly not a treifa because the goral would not choose a treifa.
2) Couldn't the Gemara prove the idea of rov from the mitzvah of yibum? There is a machlokes if a katan can do yibum, but really how can even a gadol do yibum - maybe he is unable to father a child and cannot fulfill "lehakim shem l'achiv" and is therefore pogaya b'ervah. It must be we follow the rov of people, who are able to have a child. And even if you deflect this by saying maybe yibum only applies if the yavam already has kids, so you know he is able to, there is still a possibilty that now he became unable to, so it must be based on rov. Why doesn;t the Gemara prove it from here?
Y Barach, Brooklyn NY
This is an interesting question
(1) The Pshat in "Ein Goral Kove'a l'Azazel Ella b'Davar ha'Ra'uy Lashem" is not that Hash-m makes a miracle and that the Goral automatically chooses an animal which is valid for the purpose. Rather, what it means is that there is a Halachah that even though the goat sent out into the wilderness is not going to be offered as a sacrifice, and therefore one might have thought that it does not matter if it is invalid as a Korban, nevertheless since one does not know at the time one makes the Goral which goat is going to be for Hash-m and which one is going to be sent out to the desert, therefore both must be valid as a sacrifice. This is explained by Rashi 11b DH Ein as the reason why one cannot bring a Tereifah even though it later will be thrown off the rock.
So the Gemara goes as follows: At the bottom of 11a we say that the 2 Se'irim must be equal. The Gemara asserts that the reason why we are not worried that one of them is a Tereifah (Rashi explains that this refers to the Azazel goat to be sent out to the Midbar because the one for Hash-m can be checked after slaughtering) must be because we follow the Rov. The Gemara then asks what difference does it make if the Azazel goat is a Tereifah - anyway it won't be brought as a Korban? If so we no longer have a proof that we follow the Rov? The Gemara answers that in fact the Azazel goat also must not be a Tereifah because of "Ein Goral Kove'a..." - since we do not know at the time of the Goral which one is going to be for Hash-m, therefore the Halachah is that they must both be Kosher. Since it is impossible to check the Se'ir l'Azazel after it was thrown off, it must be that we follow the majority of animals which are not Tereifah.
(2) This is quite a tricky question but I will try to cite some sources from Yevamos that may make things clearer.
(a) There are ways of checking whether an adult is capable of having children. See Yevamos 80a that the way of knowing if someone was born a "Saris" who cannot have children is to see whether when he urinates, the urine makes an arch. The Gemara there mentions a number of other symptoms. The Gemara there 80b also mentions signs of an "Ailonis" - a woman who cannot have children, for instance if she has a deep voice like a man.
You might want to argue that we might do all the tests and they might be positive but even so there might be some other reason why the person is incapable of having kids so this proves that we have to rely on the Rov. Well, I found that the Nimukei Yosef Yevamos 3b in the Rif pages writes that the reason an elderly or a barren lady may do Yibum is because we encountered several barren and old women who eventually had children, so they are considered capable "Lehakim Shem". In contrast an Ailonis possesses specific deficiencies which we know will make it impossible for her ever to have kids.
The Keren Orah Yevamos 79b DH u'MS'C expresses this by writing that anyone who does not possess a noticeable physical deficiency - like an Ailonis - is not excluded by the verse "Who will give birth" (see Devarim 25:6 and Yevamos 12a). Such a person possesses the reproductive organs and there is some other reason why she does not have children.
In contrast concerning the Katan and Ketanah mentioned by the Gemara at the beginning of our Sugya in Chulin, there is no way of knowing yet whether they will be capable of having children, so it must be that the Torah is relying on the Rov.
(b) Regarding the second point you made that even if we know the Yavam was at one time capable of having children, nevertheless it is possible he afterwards became incapable:-
See Mishnah Yevamos 79b where Rabbi Akiva stated that a person who was born healthy and became a eunuch due to an accident etc. that happened later, may do Chalitzah because he had a "Sha'as ha'Kosher" - he was originally fit for Yibum and this is sufficient. A Beraisa in the Gemara there states that even though he should not Lechatchilah do Yibum, nevertheless if he did so, this is effective. One sees from this that if a person was originally capable of performing Yibum it does not matter what happened to him later.
Yeyasher Koach and Shabbat Shalom
Dovid Bloom