Private meet budgets way more often, because there's a finite amount of money. Lenders sign term sheets detailing exactly how much money they'll lend and on what terms, so the developer has precisely that amount of money to spend. It's a big deal to need more funds, because it requires a ton of legal work, probably new banks, new agreements, new terms, etc - all of which might conflict with the existing funding structure.

So yeah private ones usually do meet budget.

Source - am architect