I stopped using Designer a while back because of this and other issues, but I love the good things about Designer and really want it to become a great app, so I come to check on it from time to time to see if any progress has been made.
I am astounded and rather disappointed that nothing has been done to address file management in the nearly three years since this thread was opened. I really like Aaron Martin's example improvements from April 2nd, 2020, and I feel like this is a relatively simple solution that would go a long way toward improving the usability of this aspect of the app. But apparently this either isn't a development priority, or Serif just isn't interested in spending development resources to improve Designer on iPad? I can't really tell.
I once deleted something I was working on because there is no indication at any point whether something is correctly saved or not before you close it. Even though I was aware of how the system does work and consciously trying not to do that!