The D'var Avrohom (2:1:4) introduces an interesting sevara. He is discussing the machlokes between the shach (ohr zarua) and darkei moshe whether a father is able to appoint someone as his agent to do mila. The Ohr Zarua and Shach hold that there is no concept of shlichus, rather if the father can't do it himself he has no choice but to forfeit the mitzvah and allow someone else to do it. The D'var Avrohom explains that the concept of shlichus only applies when the mitzvah is only on the sender but not on the agent. The action of the agent can only be attributed back to the sender when the action is not needed by the agent himself for his own mitzvah. However, anytime the agent also has a mitzvah, he cannot be considered to be doing the shlichus of the mi'shalei'ach. A theoretical example would be that even if we would allow shlichus by mitzvas that are incumbent on the individual i.e. tefillin, we wouldn't be able to consider shimon's putting on tefllin as a shaliach of reuven to be as if reuven himself put on tefillin because shimon has his own mitzvah to put on tefillin so the action is grabbed by his own mitzvah and isn't associated with reuven. Therefore, the Shach holds that when the father forfeits his mitzvah of mila either because he can't do it or because he chooses not to, the mitzvah is incumbent upon everyone (the chiyuv on beis din in this context means any Jew who is able to do it). Since the mohel is going to be doing it for his own mitzvah, the action cannot be attributed to being done as a shaliach of the father.
It seems to me that our gemara clearly rejects the sevara of the d'var avrohom. The gemara says that when a korban is sent from overseas the kohein does the waving for the person. The gemara isn't clear whether it means that the kohein does it instead of the person, or actually does it as a shaliach of the person to assist the sender in their obligation of waving. However, Rashi on 61b clearly writes that when a woman or goy send a korban the kohein serves as their "shaliach" to wave it for them (even for semicha we would have made the kohein their agent if not for a special passuk excluding the ability to appoint a shaliach for semicha). This seems to be the gemara's intent also when the korban is sent from overseas - the kohein does tenufa as the shaliach of the sender. Now, the gemara on the top of 61b quoted a contradiction in pesukim whether the kohein does tenufa or the owner does tenufa, and concludes that they do it together since there is a mitzvah on both of them. We see here that even though the mitzvah of tenufa is an obligation on the kohein himself, he cans still serve as an agent of the owner to do semicha for them. This seems to be a strong proof against the sevara of the d'var avrohom who says that when the agent (kohein in this case) has his own obligation, he cannot be an agent of someone else because the action is used for his own obligation and cannot be used for the mi'shaleiach.