The @youjin solution have an issue when you use it inside a Sheet with navigationView.

For example, first I present an .sheet view, inside this sheet view I have and NavigationView as child, then, inside all this, I present the Contact Picker. For this scenario when Contact Picker dismiss, also dismiss my sheet view parent.

I added an @Environment(\.presentationMode) variable and I dismissed using the Coordinator approach. Look my solution here:

import SwiftUI
import ContactsUI

/**
Presents a CNContactPickerViewController view modally.
- Parameters:
    - showPicker: Binding variable for presenting / dismissing the picker VC
    - onSelectContact: Use this callback for single contact selection
    - onSelectContacts: Use this callback for multiple contact selections
*/
public struct ContactPicker: UIViewControllerRepresentable {
    @Environment(\.presentationMode) var presentationMode
    
    @Binding var showPicker: Bool
    @State private var viewModel = ContactPickerViewModel()
    public var onSelectContact: ((_: CNContact) -> Void)?
    public var onSelectContacts: ((_: [CNContact]) -> Void)?
    public var onCancel: (() -> Void)?
    
    public init(showPicker: Binding<Bool>, onSelectContact: ((_: CNContact) -> Void)? = nil, onSelectContacts: ((_: [CNContact]) -> Void)? = nil, onCancel: (() -> Void)? = nil) {
        self._showPicker = showPicker
        self.onSelectContact = onSelectContact
        self.onSelectContacts = onSelectContacts
        self.onCancel = onCancel
    }
    
    public func makeUIViewController(context: UIViewControllerRepresentableContext<ContactPicker>) -> ContactPicker.UIViewControllerType {
        let dummy = _DummyViewController()
        viewModel.dummy = dummy
        return dummy
    }
    
    public func updateUIViewController(_ uiViewController: _DummyViewController, context: UIViewControllerRepresentableContext<ContactPicker>) {

        guard viewModel.dummy != nil else {
            return
        }
        
        // able to present when
        // 1. no current presented view
        // 2. current presented view is being dismissed
        let ableToPresent = viewModel.dummy.presentedViewController == nil || viewModel.dummy.presentedViewController?.isBeingDismissed == true
        
        // able to dismiss when
        // 1. cncpvc is presented
        let ableToDismiss = viewModel.vc != nil
        
        if showPicker && viewModel.vc == nil && ableToPresent {
            let pickerVC = CNContactPickerViewController()
            pickerVC.delegate = context.coordinator
            viewModel.vc = pickerVC
            viewModel.dummy.present(pickerVC, animated: true)
        } else if !showPicker && ableToDismiss {
//            viewModel.dummy.dismiss(animated: true)
            self.viewModel.vc = nil
        }
    }
    
    public func makeCoordinator() -> Coordinator {
        if self.onSelectContacts != nil {
            return MultipleSelectionCoordinator(self)
        } else {
            return SingleSelectionCoordinator(self)
        }
    }
    
    public final class SingleSelectionCoordinator: NSObject, Coordinator {
        var parent : ContactPicker
        
        init(_ parent: ContactPicker){
            self.parent = parent
        }
        
        public func contactPickerDidCancel(_ picker: CNContactPickerViewController) {
            parent.showPicker = false
            parent.onCancel?()
        }
        
        public func contactPicker(_ picker: CNContactPickerViewController, didSelect contact: CNContact) {
            parent.showPicker = false
            parent.onSelectContact?(contact)
        }
    }
    
    public final class MultipleSelectionCoordinator: NSObject, Coordinator {
        var parent : ContactPicker
        
        init(_ parent: ContactPicker){
            self.parent = parent
        }
        
        public func contactPickerDidCancel(_ picker: CNContactPickerViewController) {
            parent.showPicker = false
            parent.onCancel?()
            parent.presentationMode.wrappedValue.dismiss()
        }
        
        public func contactPicker(_ picker: CNContactPickerViewController, didSelect contacts: [CNContact]) {
            parent.showPicker = false
            parent.onSelectContacts?(contacts)
            parent.presentationMode.wrappedValue.dismiss()
        }
    }
}

class ContactPickerViewModel {
    var dummy: _DummyViewController!
    var vc: CNContactPickerViewController?
}

public protocol Coordinator: CNContactPickerDelegate {}

public class _DummyViewController: UIViewController {}

What I did, is just wrapping it inside a NavigationController. Maybe not as clean as arturigor's answer, but works quite easily.

func makeUIViewController(context: Context) -> some UIViewController {
    // needs to be wrapper in another controller. Else isn't displayed
    let navController = UINavigationController()
    let controller = CNContactPickerViewController()
    controller.delegate = delegate

    controller.predicateForEnablingContact = enablingPredicate

    navController.present(controller, animated: false, completion: nil)
    return navController
}

Regarding the questions, how it should be displayed. I Just have it displayed conditionally as a view inside a group

Group {
    Text("Sharing is caring")

    if showContactPicker {
        ContactPicker(contactType: .email)
    }
}

First of all, please file a [Bug Report][1] for this issue. [1]: https://bugreport.apple.com

Secondly, there are 2 workarounds for this issue:

1. You can use ABPeoplePickerNavigationController which is deprecated but still works.
2. Create a UIViewController which presents CNContactPickerViewController on viewWillAppear and use this newly created view controller with SwiftUI.

1. ABPeoplePickerNavigationController

import SwiftUI
import AddressBookUI

struct PeoplePicker: UIViewControllerRepresentable {
    typealias UIViewControllerType = ABPeoplePickerNavigationController

    final class Coordinator: NSObject, ABPeoplePickerNavigationControllerDelegate, UINavigationControllerDelegate {
        func peoplePickerNavigationController(_ peoplePicker: ABPeoplePickerNavigationController, didSelectPerson person: ABRecord) {
            <#selected#>
        }
        
        func peoplePickerNavigationControllerDidCancel(_ peoplePicker: ABPeoplePickerNavigationController) {
            <#cancelled#>
        }
    }

    func makeCoordinator() -> Coordinator {
        return Coordinator()
    }

    func makeUIViewController(context: UIViewControllerRepresentableContext<PeoplePicker>) -> PeoplePicker.UIViewControllerType {
        let result = UIViewControllerType()
        result.delegate = context.coordinator
        return result
    }
    
    func updateUIViewController(_ uiViewController: PeoplePicker.UIViewControllerType, context: UIViewControllerRepresentableContext<PeoplePicker>) { }

}

2. CNContactPickerViewController

EmbeddedContactPickerViewController

import Foundation
import ContactsUI
import Contacts

protocol EmbeddedContactPickerViewControllerDelegate: AnyObject {
    func embeddedContactPickerViewControllerDidCancel(_ viewController: EmbeddedContactPickerViewController)
    func embeddedContactPickerViewController(_ viewController: EmbeddedContactPickerViewController, didSelect contact: CNContact)
}

class EmbeddedContactPickerViewController: UIViewController, CNContactPickerDelegate {
    weak var delegate: EmbeddedContactPickerViewControllerDelegate?
    
    override func viewWillAppear(_ animated: Bool) {
        super.viewWillAppear(animated)
        self.open(animated: animated)
    }
    
    private func open(animated: Bool) {
        let viewController = CNContactPickerViewController()
        viewController.delegate = self
        self.present(viewController, animated: false)
    }
    
    func contactPickerDidCancel(_ picker: CNContactPickerViewController) {
        self.dismiss(animated: false) {
            self.delegate?.embeddedContactPickerViewControllerDidCancel(self)
        }
    }
    
    func contactPicker(_ picker: CNContactPickerViewController, didSelect contact: CNContact) {
        self.dismiss(animated: false) {
            self.delegate?.embeddedContactPickerViewController(self, didSelect: contact)
        }
    }
    
}

EmbeddedContactPicker

import SwiftUI
import Contacts
import Combine

struct EmbeddedContactPicker: UIViewControllerRepresentable {
    typealias UIViewControllerType = EmbeddedContactPickerViewController
    
    final class Coordinator: NSObject, EmbeddedContactPickerViewControllerDelegate {
        func embeddedContactPickerViewController(_ viewController: EmbeddedContactPickerViewController, didSelect contact: CNContact) {
            <#selected#>
        }
        
        func embeddedContactPickerViewControllerDidCancel(_ viewController: EmbeddedContactPickerViewController) {
            <#cancelled#>
        }
    }

    func makeCoordinator() -> Coordinator {
        return Coordinator()
    }

    func makeUIViewController(context: UIViewControllerRepresentableContext<EmbeddedContactPicker>) -> EmbeddedContactPicker.UIViewControllerType {
        let result = EmbeddedContactPicker.UIViewControllerType()
        result.delegate = context.coordinator
        return result
    }
    
    func updateUIViewController(_ uiViewController: EmbeddedContactPicker.UIViewControllerType, context: UIViewControllerRepresentableContext<EmbeddedContactPicker>) { }

}