1) The crucial point is what the Ramban here writes: it is "b'Yado l'Garshah." Since he could divorce her now, and she would thereby be exempt from Yibum, the level of belief that he possesses is just as strong as two witnesses when he says that she is exempt from Yibum for a different reason. The special Migu in our Gemara is just as strong as two witnesses.

2) The Ritva gives different reasons for why this Migu is just as strong as two witnesses:

a. There is a rule that "Ein Adam Chotei v'Lo Lo" (see Gemara above, 63b). A person is not suspected of sinning if he will obtain no personal benefit from his crime. Therefore, we do not suspect the husband of falsely saying that his wife does not require Yibum, since he himself will not benefit from this.

b. The Ritva writes that if he falsely says that there are sons or there are no brothers, his lie will be revealed later on. We do not suspect a person of lying, if it is easy to catch him on this falsehood.

c. Finally the Ritva states an important rule in the laws of Yibum. When a husband dies we assume that his widow has a Chazakah that she does not need to do Yibum until we actually know that there is indeed a Yavam waiting for her. This is an additional reason to believe the husband.

The Ritva concludes that because of these three reasons the Migu of the husband is just as good as two witnesses.

2) The Rashba (begining of 64a) adds explanation for why the Migu here is stronger than a standard Migu. The standard Migu is where one has two possible arguments which could be made. Both arguments may very possibly not be true. However, we believe the present argument because if he would have wanted to lie he could have said a better lie. However, such a Migu possesses an inherent lack of strength because it may always be that he is saying the weaker argument simply because he prefers it even though he has not systematically thought out which argument is more likely to be accepted.

In contrast, the Migu in our Gemara is much better, because if he wants, he could divorce her now in front of us. We can explain the Rashba to mean that the crucial difference is between merely talking or actually doing. A standard Migu works on the basis that if he wanted to lie he could have said something more plausible. There, it is all talk and it follows that the power of a Migu is limited because there might be other reasons -- that we are not aware of -- for why he said what he said. In contrast, if he could have actually done something practical to exempt his wife from Yibum (namely, by divorcing her), this means he posseses a much stronger believability when he says that she is exempt. Therefore, when he says that he has sons he is believed.

(Possibly, the words of the Rashba do not state explicilty what I have written above, but I believe that this is what he means. In addition, I think that what the Rashba wrote helps us understand better what the Ramban, who was the teacher of the Rashba, meant.)

3) Now to answer your third question: why can a Migu not work to put someone to death? The reason is that it is clear that one needs a much stronger proof than a Migu in order to put someone to death. In the Mishnah in Makos 7a, Rebbi Tarfon and Rebbi Akiva say that had they been on the Sanhedrin, no one would ever have been sentenced to death. They would have deliberately asked the witnesses questions that they were unable to answer. To sentence someone to death, a proof even stronger than the proof for allowing a woman to remarry is needed.

Furthermore, according to the Ramban and Rashba, we believe the husband with his Migu only because he could have divorced her. Accordingly, to accept a Migu in order to sentence someone to death, there would have to be the possibility to actually kill him without the sentencing, which clearly is impossible.

4) I later found a different answer to your original question, given by one of the Mefarshim.

a. Your question is based on the assumption that the Sugya is dealing with a case of Ervah. However, it is not so simple that the prohbition of a Yevamah marrying a man other than the brother-in-law is in fact considered Ervah. There is a major discussion on this, and it seems that the majority of Poskim maintain that it is Ervah, but there is a significant minority who maintain that it is not.

b. Possibly the clearest source that maintains that Yibum is Ervah is the Mordechai in Yevamos, end of #58, who writes, "The permission that a Yevamah requires in order to remarry someone other than the brother-in-law is considered as Ervah... and therefore requires two witneses." However, the Noda bi'Yehudah (Even ha'Ezer Kama, #54, DH v'Hineh Ra'isi) disagrees with the Mordechai. He writes that if the child of a prohibited union is not a Mamzer, then that prohibition is not considered Ervah. It follows that since is it is only a negative commandment, and not a more serious "Kares" prohibition, for the widow whose husband had no children to marry someone other than the brother-in-law, if she transgressed this Mitzvah the offspring is not a Mamzer. Only from a "Kares" prohibiton does a Mamzer result. The proof of the Noda bi'Yehudah that Yibum is not Ervah is from Sotah 18b, where Rav Hamnuna says that if a woman waiting to do Yibum was unfaithful and had relations with another man, she is prohibited to marry the Yavam in the same way that a married woman who was unfatihful must leave her husband. However, the Gemara there proceeds to say that the Halachah does not follow Rav Hamnuna. The Noda bi'Yehudah argues that this proves that a Yevamah is not similar to an ordinary wife and therefore Yibum is not Ervah.

c. If we say that Yibum is not Ervah this might help answer your questions. In fact, I found that the Teshuvos Zecher Yitzchak (by Rav Itzela Ponevitcher zt'l), end of #12, DH v'Im, writes,

"The Migu of Mah Li l'Shaker works (Kidushin 64b). It is only for a Davar she'b'Ervah, where witnesses are needed, that a Migu does not work."

We see that the Zecher Yitzchak understands that the reason why the Migu works in our Gemara is because Yibum is not considered Ervah. Therefore, a lower level of belief is sufficient to prove that the widow need not marry the brother-in-law. According to the Zecher Yitzchak, there is support from our Gemara for the Noda b'Yehudah.

d. Therefore, in order to answer your very valuable questions, Yaakov, we have come up with at least two important Halachos, according to some opinions:

1. According to the Ritva, when a man dies his wife has a Chazakah of being permitted to marry whomever she wants, unless we know that there is a Yavam waiting. This helps us understand why a somewhat lower level of evidence is sufficient to allow her to marry whomever she wishes.

2. The Zecher Yitzchak learns similarly that our Gemara fits well with the opinion held by the Noda b'Yehudah that Yibum is a slightly lighter prohibition and is not considered Ervah.

Kol Tuv,

Dovid Bloom