I'm a former tech consulting manager at one of the B4 and my experience is that the types of multi-million dollar software projects that they get involved in are lose-lose situations from the start. They are always high-profile, huge projects that a ton of executives, salespeople, etc all have their hands in, and everyone is fully aware that the delivered product is going to be a steaming bureacratic pile of shit with a never-ending list of feature requests and bugs, no matter who they hire. Then they hire Deloitte/IBM/Accenture because Deloitte/IBM/Accenture are so big that they are the only companies that will knowingly take on a losing project as long as they get paid handsomely for it.
And if the project is gonna be a loser anyway, why would you waste your best people on it instead of the C-stringers from your offshored development center? At my B4, the major multi-million dollar government projects like that were basically the company graveyard where you were staffed when literally no other project would take you.
For the actually successful and typically much smaller software projects, we actually had a subsidiary company with a much better reputation that would handle the design and development instead of our offshore dev centers.