Every yard has a day which is busier than the rest.
Wouldn’t it make sense to have job patterns like the following?
To achieve 24/7 cover and make most efficient use of each vehicle:
Sunday-Saturday One driver on 06:00-18:00 shift, the rotate-around “set mate” on 18:00-06:00.
Four shifts on, four off. You’d need A pair of drivers to do Sunday-Thursday, another pair to do Tuesday-Friday, another pair to do Thursday-Sunday. and the final pair to do Saturday-Tuesday.
Thus, eight drivers, four on permanent days, four on perm nights - would tie up a total of FOUR vehicles 24/7
The night shift has a built-in 25% uplift for “shift allowance”.
Any “Overtime” would be covering absent colleagues, rather than working beyond 12 hours in a shift - with shifts actually being planned for TEN hours - but given 12 - to allow for normal delays and slippage.
“Nights Out” - only in emergencies, and paid at a decent fair amount, rather than “expenses tax free”.
FIXED SHIFTS - allows a worker to plan weeks and months ahead in any aspect of their private life.
ENOUGH of the “unplannable slavery of any-five-from-seven”.
Don’t like Earlies/Nights? - No need to rotate any more, neither!
Don’t like working across weekends? - Better than working 12-15 hours 5-6-5-6 though!
That would be the only compromise here. Take it or leave it.
Rotating the actual days one works - is far better than rotating from days to nights, which’ll see you in an early grave if you do it too often… 
Money? - Is secondary to the right terms and conditions eh?