Outsourcing allows you to isolate the status page from your human infrastructure. If your team makes a bad decision that leads to an outage then they can make the same bad decision about your status page if they are in charge of that as well. This provides a form of human fault tolerance.

I agree that insulating against human fault is always a good thing to do but at some point you must trust your team to do the right thing. In fact in using a 3rd party you are trusting that said 3rd party also has a form of human fault tolerance.

That said I cannot imagine even the most incompetent team making a change to the status page, which I would always host on completely separate infrastructure, at the same time as the rest of production infrastructure.