Yoma 2a - L'hotzi mi'liban shel Tzadukim

Could be I should wait till I'm learning Chagigah, but: Here we are m'tamei the cohen that will burn the parah adumah, and tovel him. We hold that's okay, and the tzedukim hold you need to wait till nightfall. So by using such a cohen we are "removing [this opinion] from the heart of the tzadukim". How does that work exactly?

The question is especially sharp according to Tosefos ד"ה להוציא that we actually would not be metamei him tumah d'oraysa (see also 3b Tos. ד"ה ריב"ב) but only a tumah based on a d'rabanan, which the tzadukim clearly will not even notice. Of course, they may still notice that he goes to the mikveh, and worry. What does all this accomplish?

Update: I see that the Tosefos Yeshanim says something that sounds similar to my suggestion, and if so, baruch shekivanti!