Yes, but I think the missing ingredient is that building societies "bid" to decide who gets to go first. Whoever is willing to take the smallest loan/pay back fastest gets preference.
Reinventing building societies seems very cool to me -- it's a little more nuanced than the birthday thing, but it's a system that can work with assets that generate value.
You may not have £100,000 but if you get 1000 people with £100 then you can buy a house.
Obviously the birthday system doesnt address the interest, but I'm sure your friends would be ok paying interest right?