Utmost privacy? In practice, there is always risk. Replace your iPhone with a simpler feature phone, or dispense with a phone entirely. What you don’t have is more difficult for an adversary to compromise.
Seeking to feel secure with a phone or most any other complex device is laudable, but utmost security is impractical at best, nor is that even really possible, because there is always the possibility of a compromise by a sufficiently-wealthy adversary, or by an adversary with local or trusted access.
Complex devices can be vulnerable. Far simpler devices can also be vulnerable, too. Two classic and older examples:
And there are many potential exploits which do not involve an iPhone at all.
Now you can make it harder and more expensive to compromise your gear. Better physical security. Better electronic security. Better practices. For better security against the more common and less well-funded adversaries, start here:
But you will unlikely be satisfied with this or any answers. You just won’t be. Because absolute security is impossible. There is always risk.
As for broadcasting to or from your screen, that’s exceedingly unlikely. And if you're worth enough for that mirroring to happen, you’re way past the sort of security advice that can be offered in this or any forum.