refactor: Use the pubspec_parse package for parsing

[spydon]

Description

Since the pubspec package is deprecated this PR refactors and cleans up the code by using the official pubspec_parse package.

Closes: #778

Since this is a pretty big change we should maybe release it as a pre-release version first.

Type of Change

• ✨ feat -- New feature (non-breaking change which adds functionality)
• 🛠️ fix -- Bug fix (non-breaking change which fixes an issue)
• ❌ ! -- Breaking change (fix or feature that would cause existing functionality to change)
• 🧹 refactor -- Code refactor
• ✅ ci -- Build configuration change
• 📝 docs -- Documentation
• 🗑️ chore -- Chore

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[MarkOSullivan94]

@spydon I noticed your comment here

#778 (comment)

Why did you use pubspec_parse and not pubspec_manager?

[spydon]

@spydon I noticed your comment here

#778 (comment)

Why did you use pubspec_parse and not pubspec_manager?

Because we only needed it for parsing (and it's the official pubspec parsing package), and it also has a cleaner API imho.