Apple implements things which differentiates the iPhone and makes the experience better. If they see search as a way of doing that, they might well do it. There are a lot of services which are accessible to iPhone users which are completely free for years. The App Store, Find My iPhone, Find My Friends, the iCloud APIs (not the storage you buy for backup, but the API servers used by 3rd party developers). Maps is the obvious/ big one which is extremely expensive to offer.
They also offer content services like Music, TV+, and News+... services which they pay 3rd parties to use or develop content for. These are all quite distinct from Apple most other Apple services in that they are about the content someone else creates. Search definitely doesn’t fit into the same bucket as Apple Music and Apple TV+.
Apple News and the App Store itself are the only parts of Apple which get some revenue from advertising platforms.
If they see search the way they see maps, it could be free. It’s possible they will push an advertising platform with search. I don’t think it’s anywhere near a slam dunk though, and it’s possible it will have a small advertising load to support some costs like the App Store does.
Apple doesn’t need Search to be a profit center. They might put advertising on the platform (in fact I’d say it’s likely). But they don’t need it to look like Google’s pile of adverts with a few organic results at the bottom.
Apple's choice for Maps was to pay Google or some other service a per user fee to provide turn by turn directions, lose all their users to Android (which had that feature for free), or make their own maps. They didn't provide it for free to differentiate -- the experience was laughably bad at launch. They provided it for free because their cash cow was about to keel over.
I'd love it if they made Apple Music and Apple TV free too.