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.
It would be rather silly to host your status page on your main production infrastructure since if it goes down so does your status page.
Personally unless you're a major player I think you should always outsource your status page.