Revert "Revert recipe deps and path migrations."

This reverts commit 81bcf62b91d0c92da8c0998ca5b29bd41964da42.

Reason for revert: Fixed the incorrect paths.

Original change's description:
> Revert recipe deps and path migrations.
>
> Revert "Roll recipe dependencies (trivial)."
>
> This reverts commit 88e0e483710ce197191344ca60ed60be97a20419.
>
>
> Revert "Remove deprecated uses of the path subscript operator"
>
> This reverts commit d1cfc703d24ec96c6b02a5133a945f07aa43b62e.
>
>
> Revert "Use parent in path.parents"
>
> This reverts commit 3441020b7e5ad25807bd0a289357e65f48401ad8.
>
>
> Revert "Replace deprecated path joins with / operator"
>
> This reverts commit 57f92aba78607ab84cae8db1458b8461b95cf8f5.
>
> Bug: https://github.com/flutter/flutter/issues/154025
> Change-Id: Ic4405aefb8eeef4713eb56a1099527206348d938
> Reviewed-on: https://flutter-review.googlesource.com/c/recipes/+/59440
> Reviewed-by: Zach Anderson <zra@google.com>
> Commit-Queue: Christopher Fujino <fujino@google.com>

# Not skipping CQ checks because original CL landed > 1 day ago.

Bug: https://github.com/flutter/flutter/issues/154025
Change-Id: I75fdd4f0c2b6cf7dbe7da3c32881a9d4c69e4620
Reviewed-on: https://flutter-review.googlesource.com/c/recipes/+/59460
Reviewed-by: Zach Anderson <zra@google.com>
Commit-Queue: Alexander Thomas <athom@google.com>