When the Torah uses the term "finger" or the term "kohen," it is understood to always mean the right hand. The bias against the left-hand is already understood, but note that the Torah specifies the right-hand on occasion. But there are other occasions when the left-hand was explicitly called for (also as per the Torah) - to the extent that Rabbi Shimon allowed left-handed acceptance of the blood. Going back to the verses, "finger" AND "kohen" is necessary in his estimation. And the mishnah seems not to have included this machloket.