Men have priority over women - even for lost objects

[updated with Mishna] Rav Sternbuch(Teshuvos v’Hanhagos 4:217): In the Mishna at the end of Horios it is taught, A man take priority over a woman in regard to returning lost objects. The reason for this is because the man has a greater obligation to do mitzvos than a woman. We also find in Shulchan Aruch concerning tzedaka that a man comes before a woman. This is very strange. I can understand that a Torah scholar takes precedence to a woman because it is showing respect for the Torah. However there is no obligation to honor every man because of the mitzvos that he fulfills. I can understand that in saving of life that a man comes before a woman because he can serve G‑d more since he has much more mitvos and thus is sanctified more. However what does that have to do with returning him his lost objects before returning a woman her lost objects – there simply isn’t a general obligation of honoring each man? I could not find this din in the Shulcha Aruch or the Rambam, but it is explicit and clear in the Mishna. This question requires additional study. Perhaps you can answer that since the man is obligated in more mitzvos, when you give priority to the man you show with this the importance of mitzvos. Therefore even though there is no requirement to honor him, but in a situation where you have both a man a woman that you need to returnsomething, we give priority to him because he is obligated in more mitzvos. With this we cherish and show the importance of mitzvos. Therefore the Sages required in such a case to give preference to the man. However this still requires clarification.
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Horious(13a): Mishna: A man takes precedence over a woman in life and death matters and the return of lost objects [ because of greater holiness resulting from having more mitzvos]. A woman takes precedence over a man regarding being clothed [because she suffers more from embarrassment and shame of being naked] and redemption from captivity. If they are both faced with rape then a man takes precedence over a woman [because rape is unnatural for a man and thus more degrading than for a woman].