So I went to visit my old yeshiva this past weekend, and I got to hear lots of interesting vorts. Here’s one of them:
At the Friday night oneg, one of my rabbeim asked why the me’il did not have tzitzit on it. In typical sacrilegious fashion I called out, “maybe because it was a טלית שכולה תכלת”! After laughing off my comment, my rebbi asked, “And who says it didn’t have tzizit? The picture books say so.” Even so, he went on to say a great peshat, in the name of the Beit Yitzḥak:
The Gemara in Arakhin says:
הכל חייבין בציצית, כהנים לוים וישראלים: פשיטא! כהנים איצטריך לי', סד"א הואיל וכתיב לא תלבש שעטנז – גדילים תעשה לך, מאן דלא אישתרי כלאים לגביה בלבישה הוא דמחייב במצות ציצית, והני כהנים הואיל ואשתרי כלאים לגבייהו לא לחייבו, קמ"ל נהי דאישתרי בעידן עבודה בלא עידן עבודה לא אישתרי.
[It was taught:] Everyone is obligated in tzitzit; priests, Levites, and Israelites. This is obvious! It was necessary to teach priests, because I would have thought that since it says “You shall not wear sha’atnez” next to “Fringes you shall make for yourself” – only one who is not allowed to wear sha’atnez is obligated in the commandment of tzitzit, but these priests, since sha’atnez is permitted for them [in the בגדי כהונה], perhaps they are not obligated! Therefore it was necessary to teach that they are, [and the reasoning is] for although during the service they are permitted [to wear sha’atnez], any other times they are not.
Generally one would understand the Gemara as saying that the fact that priests are not always permitted to wear sha’atnez is a reason to dismiss the derivation we were trying to make from the juxtaposition of the two commandments. Says the Beit Yitzḥak, this is incorrect. We do indeed derive this halakha, but it is limited to the specific times that the priestly garments are being worn! Therefore, said the rebbi, it makes perfect sense why the picture books show the me’il without tzitzit!
 It should be noted that is questionable if the me’il was actually a kind of garment that would require tzitzit; see Ramban to Exodus 28:31.
 See Rashi to Numbers 16:1 s.v. ודתן.
 Deuteronomy 22:11
 Arakhin 3a