You yourself agree that these Korbanos come only from communal funds. A volunteer guard would compromise that communal character at the very first link in the procurement chain — before the grain even enters the lishka. The halacha follows the Chachamim.

The R' Yose case is just one instance of a larger live dispute: could a wealthy individual ever sponsor the Omer, the Shtei HaLechem, or the Lechem HaPanim directly — even willingly, even generously? The Chachamim's answer is absolute: no. The communal character is not a formality but a legal requirement built into the very nature of these Korbanos. An individual's contribution — however large — cannot purchase the "communal" quality. This is why the sefichim guards must be paid from the lishka: the public ownership must be unbroken from field to altar. The Mishnah uses the guard's case precisely because it involves the Omer and Shtei HaLechem specifically — these are the Korbanos where the question of private sponsorship has the most consequence.