Even though Apple hasn’t been sued for Apple Maps, they are not safe from such antitrust charges. The antitrust cases move slowly so it’s possible that they just haven’t gotten to there.
So far, Company + Map hasn't caused any problems with even bigger players and tighter interegration (Google + Google Maps and Microsoft + Nokia ( https://news.microsoft.com/2013/09/03/microsoft-to-acquire-n... )).
Just tried it and Chrome isn't supported!
Apple would not need to pay anyone to make their search (or maps or browser) the default. It would be the default on Apple devices and not used much outside of that.
Would that bother Apple? Their platform is the most valuable to advertise on. They’ll make money. The question is, will it make as much as Google was paying?