Issue when displaying a modal after dismissing another modal

Description

Before RN 0.76.5, on iOS it was possible to show a modal and after dismissing it, showing another one.
Now, when doing the exact same thing on the old architecture the second modal is not showing up.
https://github.com/facebook/react-native/issues/47694
https://github.com/facebook/react-native/issues/48245
https://github.com/facebook/react-native/issues/48559

The error is not happening on new architecture

Steps to reproduce

1. Create two different modals
2. Show the first one by clicking on a button
3. Then, create a button that will dismiss the first modal and display the second one
4. The second one won't show up

React Native Version

0.76.5

Affected Platforms

Runtime - iOS

Output of npx react-native info

System:
  OS: macOS 15.2
  CPU: (10) arm64 Apple M1 Max
  Memory: 482.95 MB / 32.00 GB
  Shell:
    version: "5.9"
    path: /bin/zsh
Binaries:
  Node:
    version: 22.12.0
    path: ~/.nvm/versions/node/v22.12.0/bin/node
  Yarn: Not Found
  npm:
    version: 8.19.4
    path: ~/code/yonitou/hygo/node_modules/.bin/npm
  Watchman: Not Found
Managers:
  CocoaPods:
    version: 1.11.2
    path: /Users/yonitouboul/.rbenv/shims/pod
SDKs:
  iOS SDK:
    Platforms:
      - DriverKit 24.2
      - iOS 18.2
      - macOS 15.2
      - tvOS 18.2
      - visionOS 2.2
      - watchOS 11.2
  Android SDK: Not Found
IDEs:
  Android Studio: 2024.1 AI-241.18034.62.2411.12071903
  Xcode:
    version: 16.2/16C5032a
    path: /usr/bin/xcodebuild
Languages:
  Java:
    version: 17.0.1
    path: /usr/bin/javac
  Ruby:
    version: 2.7.4
    path: /Users/yonitouboul/.rbenv/shims/ruby
npmPackages:
  "@react-native-community/cli": Not Found
  react: Not Found
  react-native: Not Found
  react-native-macos: Not Found
npmGlobalPackages:
  "*react-native*": Not Found
Android:
  hermesEnabled: Not found
  newArchEnabled: Not found
iOS:
  hermesEnabled: Not found
  newArchEnabled: Not found

### Stacktrace or Logs

```text
No errors logs

Reproducer

https://github.com/yonitou/modal-bug

Screenshots and Videos

When I click on the button "Close modal", it should dismiss the red one and show the blue one but the blue one never shows.
You will notice that if enabling new architecture, the problem disappears
https://github.com/user-attachments/assets/72aca692-d90d-40f7-b500-ef574ae26658

Start by running the linked yonitou/modal-bug reproducer with the old and new architectures, following the two-modal dismissal steps. Then trace the React Native iOS modal presentation and dismissal lifecycle to identify where the second modal is lost. Done means the second modal appears after dismissing the first on iOS with the old architecture, without regressing the new architecture.