Just how do gedolay hador spend their yom tov days? Well, I can’t speak for all gedolay hador – nor am I qualified to decide who is or is not a gadol – but there is a gemara that shows us how at least some gedolim behaved. And it’s not what you’d expect.
סוכה מז
אמר רב יוסף נקוט דר’ יוחנן בידך דרב הונא בר ביזנא וכל גדולי הדור איקלעו בסוכה בשמיני ספק שביעי מיתב הוו יתבי ברוכי לא בריכי ודלמא סבירא להו כמ”ד כיון שבירך יו”ט ראשון שוב אינו מברך גמירי דמאפר אתו
Rav Yosef said: take Rabbi Yochanan’s (opinion) in hand, for Rav Huna bar Bizna and all the gedolay hador went to a succah on the eighth day (of Succos outside of Israel) which might (because of ספיקא דיומא) have been the seventh day. They sat (in the succah) but they didn’t make a bracha (לישב בסוכה). Perhaps (you could argue) they held like you only make that bracha on the first day (of the chag)? I (i.e., Rav Yosef) heard (from my teachers) that (the gedolim) had come from the field.
רש”י
גמירי דמאפר קאתו. למדתי מרבותי שהן באו מן האפר שבהמותיהן רועות שם ולא ישבו בסוכה כל ימי החג
“I heard that they had come from the field” I learned from my teachers that they had come from the field where their cattle had been grazing. They therefore had not sat in a succah all through the chag (until now).
So, for the first seven days of Succos, the gedolay hador were so busy tending to their cattle that they hadn’t managed to even once sit in a kosher succah. According to Tosafos, they weren’t even able to build themselves a סוכת רועים!