I have several Experience builder apps that are built around the edit widget operating in attribute only mode, with custom settings from the Field maps designer.
Starting sometime between Mar 7 and Mar 28, the Submit button in the edit widget window sometimes will go missing. Republishing the app will usually get the button to reappear but I can't keep republishing all my apps multiple times a day- am I missing something?
I encountered a similar issue like yours.
I guess the reason is that the edit in a controller does not load until it's opened.
You can try select the edit in the controller's "Open widgets on loading":
As a follow up, after being a problem for a couple weeks the issue seemed to go away for me a few hours after making the initial post.