As a matter of fact, there are such kollelim in EVERY SINGLE town in Israel. The point of such a kollel has nothing to do with being a nistar. Its about torah in the most appropriate hour. Almost every parsha in Zohar speaks of the importance of learning torah from Chatzos and on. These kind of Kollelim (paid or unpaid) are like “oichhel peiroseihem b’oilam hazeh”. Personally, i work from 9-5 every day and cannot learn when i come home from work. So i go to sleep after spending some time with the W&Ks.
The average adult sleeps 6 hours. I tend to come home home from work at 5:30 go to sleep at 7:30 and get up at 1:30. That gives me more than 6 hours of learning time besides for Vasikin (a whole inyan in itself)!
In addition, no phone calls, no errands and other interruptions etc.
If people would just calculate the bitul zman that goes on at night, it would be an obvious choice. The most rewarding choice.