Could the anshei beit av wash their clothes and cut their hair on Thursday?

Question

Taanis (2:7):

אַנְשֵׁי מִשְׁמָר מֻתָּרִים לִשְׁתּוֹת יַיִן בַּלֵּילוֹת, אֲבָל לֹא בַיָּמִים. וְאַנְשֵׁי בֵית אָב, לֹא בַיּוֹם וְלֹא בַלָּיְלָה. אַנְשֵׁי מִשְׁמָר וְאַנְשֵׁי מַעֲמָד אֲסוּרִין מִלְּסַפֵּר וּמִלְּכַבֵּס, וּבַחֲמִישִׁי מֻתָּרִין מִפְּנֵי כְבוֹד הַשַּׁבָּת:

The kohanim of that week (but not that day) may drink wine at night but not by day (because there may be too much work by day, and they’ll need to help the kohanim appointed for that day). The kohanim of that day may not drink wine even at night (because they work at night as well). The kohanim of that week may not cut their hair or wash their clothes (so that they will do so before their week starts), but on Thursday they may because of the honor of shabbos.

The mishna doesn’t say if the daily kohanim (of Thursday) can cut their hair and wash their clothes on Thursday or not. On the one hand, they have stricter halachos (like by wine), and from the fact that they aren’t mentioned with the others as muttar, so they wouldn’t be allowed to, but on the other hand, “they may because of the honor of shabbos” applies to them to, so the would be allowed to.

So is a “Thursday kohen” allowed to wash his clothes and cut his hair or not?

Answer 1

זיל בתר טעמא (follow the logical reasons). That the אנשי בית אב can't drink wine by day or by night is because they're on duty that day; that the אנשי משמר ואנשי מעמד can't cut their hair or launder their clothes is so that they don't enter into their week of duty looking unkempt. That latter reason applies equally well to all of the בתי אבות of that week's mishmar.

(Of course, if a member of Thursday's בית אב is actually serving in the Beis Hamikdash nonstop throughout the day, he won't have the time to get his hair cut or launder his clothes, but that's a practical consideration, not a halachic one, aside from the general issur to abandon the avodah to do something else.)