Rather than trying to manage access at the various places people might be able to see the posts, it’s probably safest to create custom user roles. For example, WP autogenerates RSS feeds, so if you’re only covering the single-post and archive views, you may not be covering the feeds themselves.
If you create custom roles, you can also create custom post types. In your case, you would create bronze_member, silver_member, and gold_member roles. You can then create bronze_post, silver_post, and gold_post custom post types with rewrites so they can all appear at whatever URLs you wish. You’d need to build a custom archive template that would query all of the post types, and that way, all posts can appear together, but those who are logged in with – say – silver status will only see the bronze and silver posts, not the gold posts.
See the WP Developer References for user roles and custom post types.