There is no way to hide the "make it my default" in the standard dialog box. However, it is an idea, so go vote it up. https://community.tableau.com/ideas/5854
And for hiding stuff, is it an option for you to add tags to the URL?
And within permissions, there is an option to turn off share of customized views.
Thanks Jeff. I am looking into the second option to add tags to the URL. That sounds like a feasible option.