I just set chateau on my test site, and I have a header tab and can change headers in both Safari and in Firefox.
1. Go to users > all users and see what role is shows for your username. Only administrators can access and change themes/header/backgrounds and such.
2. Go to appearance > themes and click the "customize" button where it shows the theme you are using.
3. Temporarily switch to a different theme. Then switch back to Chateau and see if the header tab comes back under "appearance."
4. Try a different theme. Chrome has been a serious pain the last 6 months and I've actually made all my clients switch away from it as I got tired of the emails and phone calls telling me they couldn't do this or that after a Chrome update. Humor me and try a different browser. If it works, we know it is a Chrome issue and if it does not, then it is something else and staff will have to look into it.