The question arises, if the keys of rain and Techias Hamasim are not given to a messenger than how did Eliyahu perform his miracles of holding back the rain and reviving the dead child. Rashi answers that the keys can be individually given out and Tos answers that they can be given but not forever.
I was wondering if one could suggest that the chiluk in the answers depends on the chakirah in ones view of shlichus. Was is the role of the shliach? Is he an extended hand performing for the sender or does he take the kochos himself and be "Yotsei" the sender. These keys are at the roots at life. We can not relate to Hash-m by being messengers in these matters because they are beyond us.
Maybe Tos. would say we are simply an extended hand when we take "keys". Therefore, one can take 1 since he is not taking it but using it. Therefore Tos. says for a time it can be used because no Kinyan is created. Rashi on the other hand holds a full kinyan is made in shilchus. We take over the role. Therefore to an extent we can take 1 key but to take all of them is beyond us. What does the Rav think the split in answers is based upon? I couldn't find mephorshim that speak on it.
Your approach is thoughtful, but it seems difficult to say that according to Rashi, when a Shaliach receives one of the keys he becomes "Shaliach Shel Adam k'Moso." Also, if this is true, it should apply equally when the Shaliach takes all three keys as well.
In truth, though, Tosfos' answer is certainly more straightforward. Indeed, Rashi himself gives that answer in Maseches Sanhedrin. What Rashi writes here is very difficult, for the continuation of the Sugya contradicts him (as the Gevuras Ari points out) -- according to Rashi, what is the difficulty from the key of Parnasah? Moreover, Rebbi Yochanan's words do not imply at all that that the keys were not given "to one Shaliach at one time." In addition, from where did Rashi know that the keys are not given to one Shaliach at one time ? If Rashi was bothered by Tosfos' question, then he should have answered that the keys are not even given to several Sheluchim at one time, since the main point is that they are not given to a Shaliach all at one time .
Indeed, many Acharonim (see, for example, the Maharitz Chiyus) point out that perhaps the text of Rashi needs to be amended from "Yachad" to "Yichud" -- "[the keys are not given] to one Shaliach b'Yichud ," which means that they are only given temporarily, and Rashi is given the same answer as Tosfos.
According to the text that we have in our Gemaras ("Yachad"), we have to say that Rashi holds that these three keys are actually just three parts of a single key . That key is the key of life, which includes rain, Techiyas ha'Mesim, childbirth, and Parnasah (see insights to 6b). This would answer all of the questions above, and it is in this respect that Tosfos argues with Rashi and holds that there are actually three separate keys (this would also answer the question of Tosfos from the "Sar Shel Barad" and "Shel Pur'anos").