For those who don’t take the admonition of “V’hagesa bo yomam valaylah” literally, it means the reality that there are yom tovim, which require preparation time everyone from the Rosh Yeshiva, the many rabbonim who may be magid shiiur down to the workers in the chadar ochel to the janitors who sweep the floors in the beis medrash. These same people have families and require time off for rest and vacation The reality is that even the most dedicated masmid needs a break. You cannot shteig 24x7x365.
The Yeshiva system was not set up with holidays simply so that the staff could prepare for Yomim Tovim. The real reason, as you acknowledge at the end of your post – The reality is that even the most dedicated masmid needs a break. You cannot shteig 24x7x365. – Or, to phrase it slightly differently, to ensure optimal shteiging 24x7x365, humans need breaks. For some amazing bochurim – and they are not so rare, b”H – bein hazmanim means learning in a different Beis Hamedrash, perhaps browsing through some Achronim they wouldn’t during zman. I have a chavrusa who learns 12 hours a day during Bein Hazmanim. Most of us need time to recharge our batteries. This is a situation of ביטולו זהו קיומו. So, in summary, B”H is not holiday from Avoidas Hashem, but Avoidas Hashem is to holiday.