>> So I want emails to go to the SENT folder, but not be displayed on the phone.
Note that if you turn off sync for the Sent folder, as was suggested here, then the app won't be able to sync sent messages either way.
It will only store them locally, it'll look almost the same, but the app's [ Sent ] folder won't sync with anything.
Same for moving to Deleted (and Drafts too).
What I can suggest instead is:
- App settings -> account list -> expand accounts -> none
- Same screen -> turn on "inbox on account tap"