Hello Rabbi,
According to Rashi, the Talmud there is speaking about saving her from strangulation:
"Because of his sister's daughter — and she is his wife, and perhaps she will commit adultery under him, and he will have pity on her so that she not be strangled, and he writes her a bill of divorce without a date and gives it to her, and when they testify against her in religious court she produces her bill of divorce and says: I was divorced and unmarried at that time."
Does the Rabbi think Tosafot explains the Talmud there as referring to monetary law and not to saving her from strangulation?

You are right. I was mistaken. It does indeed apparently refer to covering for her so she can escape strangulation. What Tosafot says is that if there were no date on the bill of divorce, a person would cover for her and claim that she was already divorced. That implies that without his claim we would execute her by strangulation.
Tosafot adds that in such a case we would establish her on her original-status presumption, that she had relations as a married woman. Meaning that because of that, without the cover-up we would execute her due to the presumption.
Tosafot adds that apparently we should establish her on the presumption of fitness, and there is also a current presumption that she is divorced. That implies that because of these we would not execute her, so why is a cover-up needed at all?
Apparently it seems that only together with the cover-up claim would those two presumptions help so that we would not kill her. But without the husband covering for her, we would execute her because of the presumption that she was a married woman, even though those two presumptions stand against it. And that is even more puzzling. Especially since Tosafot does not explain why.
These things are very puzzling in themselves. How can one execute a woman on the basis of a formal principle when in practice we are in great doubt whether she committed adultery at all, both because of the two presumptions mentioned above, and because even without them all we have is an original-status presumption, which is a formal principle? There is no way that this is actually the law.
Suppose the woman herself does not remember whether she was already divorced or not, and out of doubt we would instruct her not to have relations because she has the presumption of being a married woman. That I am willing to accept. More than that, if she had relations in such a situation, then I have no problem with saying that she would indeed be executed. Here that is possible, because when she committed adultery she already knew that the law is that she is liable to death (even though that law is only because of an original-status presumption), and with that understanding she committed adultery. But if she had sexual relations and perhaps she was divorced, and we want to execute her by force of presumption despite the doubt — there is no way that this is the law.
Beyond that, it is not clear what exactly the case is. If there were witnesses who warned her (without that she is not liable to death), then she answered: yes, and on that condition I am doing it. Without that there is no death penalty. But if she answered that, then it is actually proven that she committed adultery as a married woman, and it is perfectly proper to execute her. It is not even clear whether a cover-up would help in such a case. And still, were it not for the presumption, we would not execute her out of concern that perhaps she was already unmarried.
Bottom line, the only way to understand this is to assume that there was warning here and that she accepted the warning, and in such a case she would be executed were it not for the cover-up. But that would not be on the basis of presumption by itself. Rather, without the presumption perhaps we would not execute her, because her very acceptance of the warning is not enough on its own for us to execute her.
If so, we have gained the conclusion that there is no death penalty on the basis of an original-status presumption. Quite apart from how Tosafot is to be understood, I do not see how one can disagree with that. Fine — on the basis of actual evidence like a majority, you can say there is evidence and therefore we execute her. But an original-status presumption is a formal rule, and the factual doubt remains in place (in addition to the two presumptions). There is no possibility that we would execute her in such a situation were it not for her acceptance of the warning.

Thank you very much, Rabbi.