How about a bit of critical thinking before yelling "Conspiracy!"? Concerning "sources", if Apple and Google are engaged in joint anti-competitive behavior, they'd probably not write press releases about it. But of course there is well-documented precedence of them doing exactly that, being caught and getting a mild slap on the wrist, so it's not a completely outlandish notion to entertain.

As for the officially stated reason: do you really think that there is an alternative search engine provider that Apple would switch to if Google didn't pay them 10 figures a year? If so, which one(s)? If not, what are they actually paying for? To mitigate the risk of a second Apple Maps? But is the case for entering the search engine market really remotely as compelling as entering the mapping space?

I'm not saying it's an open and shut case, but the idea that Google essentially pays Apple money a) as a "good-will gesture" b) to give regulators the impression that the search engine market is more competitive than it is seems possible to me. Whereas one of the alternative scenarios proposed here, that Apple would pick duckduckgo otherwise, really does not.

Personally, if I were Sundar, I'd be way more worried about anti-trust or Apple siding too much with the privacy of their users or stepping otherwise on my toes than say duckduckduckgo becoming a serious competitor because Apple anointed them default search provider after I failed to fork over enough money. I'd probably even do my best to keep a bunch of minimally viable competitors around -- not viable enough to ever pose a threat, but viable enough to keep anti-trust at bay for a bit longer. And both duckduckgo and firefox seem to fit that bill perfectly.