Fixes [Support search box inside large title nav bar](flutter#18103)
Part of [Support segmented controls in nav bars and double row nav bars](flutter#10469)

## None mode (Current default)

https://github.com/user-attachments/assets/d798314e-940f-4311-9a9a-fe999c65f280

## Always mode

https://github.com/user-attachments/assets/950a85aa-8ca2-42ea-bf8b-3cb8f95e616e

## Automatic mode

https://github.com/user-attachments/assets/c7c7240b-d493-4036-a987-30f61d02bac3

## With CupertinoSlidingSegmentedControl

https://github.com/user-attachments/assets/59f4aec4-8d9c-4223-915b-97b73cb25dc8

<details>
<summary>Sample Code</summary>

```dart
// Copyright 2014 The Flutter Authors. All rights reserved.
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.

import 'package:flutter/cupertino.dart';

/// Flutter code sample for [CupertinoSliverNavigationBar].

void main() => runApp(const SliverNavBarApp());

class SliverNavBarApp extends StatelessWidget {
  const SliverNavBarApp({super.key});

  @OverRide
  Widget build(BuildContext context) {
    return const CupertinoApp(
      theme: CupertinoThemeData(brightness: Brightness.light),
      home: SliverNavBarExample(),
    );
  }
}

class SliverNavBarExample extends StatelessWidget {
  const SliverNavBarExample({super.key});

  @OverRide
  Widget build(BuildContext context) {
    return const CupertinoPageScaffold(
      // A ScrollView that creates custom scroll effects using slivers.
      child: SafeArea(
        child: CustomScrollView(
          // A list of sliver widgets.
          slivers: <Widget>[
            CupertinoSliverNavigationBar(
              leading: SizedBox(
                width: 100,
                child: Row(
                  children: [
                    Icon(CupertinoIcons.back),
                    Text(
                      'Lists',
                      style: TextStyle(color: CupertinoColors.activeBlue),
                    ),
                  ],
                ),
              ),
              trailing: Icon(CupertinoIcons.plus),
              largeTitle: Text('iPhone'),
              // Change to desired mode.
              drawerMode: NavigationDrawerMode.none,
              drawer: Padding(
                padding: EdgeInsets.fromLTRB(8.0, 0.0, 8.0, 16.0),
                child: CupertinoSearchTextField(),
              ),
            ),
            SliverFillRemaining(
              child: Column(
                mainAxisAlignment: MainAxisAlignment.spaceEvenly,
                children: <Widget>[
                  Text('Drag me up', textAlign: TextAlign.center),
                ],
              ),
            ),
          ],
        ),
      ),
    );
  }
}
```

</details>

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