Revert "Enable lints `library_private_types_in_public_api`, `sort_child_properties_last` and `use_key_in_widget_constructors`" (#5691)

This reverts commit b1d80e0ff6e1c590fb5a01e8191fd1221ab72dd7.

This includes a fix for a latent bug in the version-check repo tooling command that caused it to fail when reverting a package that previously had a NEXT section, so that tests will pass.