Trim down dependencies for package:unified_analytics

[eliasyishak]

We should limit how many dependencies we use where possible

[eliasyishak]

Use a top level function to take in just the homeDirectory (Directory) object and use the .childDirectory and .childFile methods to fetch the path to the files... this removes the dependency on package:path

Context:

• https://github.com/dart-lang/tools/pull/247#discussion_r1525465920

[christopherfujino]

@parlough if we're still using package:file, we're not really getting rid of the dependency on package:path, as file depends on path: https://github.com/google/file.dart/blob/master/packages/file/pubspec.yaml

That's not to say we shouldn't do your change, but I'm thinking maybe we should strategize more big picture how to actually reduce dependencies in this package.