chore: playground restructure

[BobbieGoede]

🔗 Linked issue

❓ Type of change

• 📖 Documentation (updates to the documentation, readme or JSdoc annotations)
• 🐞 Bug fix (a non-breaking change that fixes an issue)
• 👌 Enhancement (improving an existing functionality like performance)
• ✨ New feature (a non-breaking change that adds functionality)
• 🧹 Chore (updates to the build process or auxiliary tools and libraries)
• ⚠️ Breaking change (fix or feature that would cause existing functionality to change)

📚 Description

Restructures playground to use Nuxt 4's structure and changes the templates a bit and removes some of the logging.

📝 Checklist

• I have linked an issue or discussion.
• I have added tests (if possible).
• I have updated the documentation accordingly.