If you search the forums, you will find that this problem has come up over and over for years with no satisfactory resolution:
You do not have permission to preview drafts.
Administrators obviously have permission to preview drafts and should never see this error message, but they do. The error message is obviously a mistake and indicates a deeper problem that actually has nothing to do with permissions, roles, and capabilities.
The problem happens if, and only if, the WordPress Address and the Site Address are different on the General Settings page. There are legitimate reasons to make these two URLs be different, such as using a shared SSL certificate, as I have already describe here.
You can argue that this is "not a core bug" if you like, in which case I will reply that this is a "design flaw" in WordPress. Whatever you call it, it is a problem, and it should be fixed. The only way to fix it to make WordPress support the kind of functionality people say they want is by changing the WordPress core.
Is anyone here interested in actually solving this problem?