Remove references to GITLAB_API_PRIVATE_TOKEN
After writing https://gitlab.com/gitlab-org/release-tools/issues/399, I noticed that this is basically a legacy environment variable, along with two others, that we should deprecate and remove for simplicity's sake.
This MR also adds the concept of a "preflight check", which we'll probably expand upon for https://gitlab.com/gitlab-org/release-tools/issues/398.