I think the proof of the opposite is true. Microsoft had Bing as the default in the United States and IE as the default browser, but it did not stop people from downloading Chrome nor setting Google as the default. The argument is that Google has the dominant position because they are the default, but I think the evidence shows that it would be the dominant search engine either way.
However, if someone built a sufficiently better one, people would switch. While they might not give up Gmail, there is very little that makes it sticky (inertia - admittedly not nothing, but not that hard to overcome given its lack of easily accessible settings).
Again, the Microsoft example shows that even if Apple picked a different default, people would pick Google on their own.
I think the flip side is true: Google is terrified of people leaving so that they pay for something that they could get for free. If Apple built a better search engine, even if they made it the default, people would switch right up until the felt it was better than Google. The important thing is that it would need to be better than Google to get people to stay.
Maps shows this as well. Despite Apple offering no choice of mapping provider, people still downloaded Google Maps Right up until Maps got good enough to get people not to bother.
From what has need described, it is not a non-compete, just Google paying for even more placement on Apple’s devices. Nothing that has been shown so far indicates that Apple was (or is) building a search engine. Nor has anything been shown that prevents them from doing so. They could be building a search engine while taking this money and when they feel they have something competitive, they could release it.