Just to check - when you upload...
You click Server -> Tableau Public -> Save to Tableau Public
Then the workbook packages, your browser opens and you see the viz.
At this point you press Edit Details
You amend these (as you need)
and click Save
Assuming you do that, I don't see anything wrong... its what I do, and mine works.
Couple of thoughts -
Do you users view the viz on public via web ... do they login? and would they login as you? (I wonder if they can download as they have 'author' permissions)
Otherwise, I don't know... you may want to check with Tableau support your public account is set to premium, if its not, you can't restrict the download of workbooks. (it should be as everyone is mean to be premium but worth a check)
as a final FYI...
I hide the control buttons etc by embedding the viz in a iframe (I'm mirroring the Tableau public site/ viz on our company website) , the dashboard/iframe is sized exactly just to cut off the bottom menu... a previous discussion about it here >> Removing functions from toolbar in embedded visualization (Public Premium)