Mishnah Parah 3:7 (with translation/explanation from Sefaria):
וּמְטַמְּאִים הָיוּ אֶת הַכֹּהֵן הַשּׂוֹרֵף אֶת הַפָּרָה, מִפְּנֵי הַצְּדוֹקִים, שֶׁלֹּא יִהְיוּ אוֹמְרִים, בִּמְעֹרְבֵי שֶׁמֶשׁ הָיְתָה נַעֲשֵׂית
And they would make the priest who would burn the cow ritually impure because of the Saducees, so they would not say, "it was done by one for whom the sun had set." [There are various forms of ritual impurity for which to become totally pure, one must immerse in water and then wait until the next sunset. The Saducees believed that the priest who burnt the red cow needed to be totally pure such that he'd only be allowed to do it if he waited for sunset. The sages held that he is able to burn the cow, even after only immersing in water, thus they made him impure and had him immerse so that everyone could see that they were not following the opinion of the Saducees.]
As a matter of fact, since the Kohen involved was rendered impure to a certain extent, the Sages enacted many other stringencies in the purity required for the preparation of the Parah Adumah ashes, so that people would not take it lightly. (See Rambam Hilchot Parah Adumah 2:1.)
What I'm trying to understand is why the Sages chose this hill to die on. The preparation of the Parah Adumah ashes is reported to have only happened seven or nine times in history (see Mishnah Parah 3:5). What would be so terrible if people mistakenly adopted the stricter view of the Saducees in this case? What would be the chances of a Parah Adumah naturally ever being prepared by a tevul yom (someone who has immersed and is waiting for sunset), and thus serving as a source of conflict between those following the views of the Rabbis and those following the Saducees?
Was this solely about discrediting the Saducees whenever and wherever possible, or is there something else going on?