Not sure what budget has to do with it, though. You're assuming that money was the bottleneck, when we don't have any proof of that, AFAIK. Could be they gave everyone all the money they need and they produced poor results regardless, a la Colonial Marines.
What? Budgets always have something to do with it, they're a business. With a bigger budget, they could have hired more software engineers earlier and fixed these problems during development rather than post-release.
That being said, and Colonial Marines is an example of this, you also need to make sure that the money being spent is being spent well, especially if you are contracting out the work to a different company.
Do they even make games? Every project they've released has had large portions outsourced.
I mean, I'm pretty sure they did almost all of the Halo 4 campaign, and Spartan Ops, and Halo 5 (which it sounds like has much more if not all of the multiplayer in-house now). Outsourcing isn't unique to 343i at all but I think MCC is way above the usual amount for AAA (correct me if I'm wrong).
I think they probably though eh, this project is relatively easy, we can just let all these other studios work on it so we can concentrate internally on Halo 5.