What I will say, however, is that the public sector has to follow a LOT of byzantine rules around procurement
They do, and it's precisely *because* they have to be scrupulously fair that there are these byzantine rules. Such rules are what happen when you attempt to codify every aspect of a transaction so that there is no conceivable way that any party can be favored by the purchaser, etc. And of course, when someone *does* find a way to do something bad within the rules, the answer is more rules.
Fewer rules means trusting human judgement. Trust in human judgment means that there *will* be the occasional misuse of funds. And misuse of funds means that someone will be blamed for not having rules to prevent the problem in the first place. (The idea that the cost of preventing a crisis can easily be more than the crisis itself something few voters can understand.)
So in the end, we get exactly what the public demands - enough rules that compliance doubles the cost of everything, but a level of fraud and graft (which might have added 10% to the price) that is much lower than the private sector.