The list of archived timeframes can get pretty long and since not active, really doesn't need to appear so hiding them would be better
as per email, leaving a long list of outdated information with no way to manage or hide it is inconvenient and increases the chances of errors (users accidentally clicking on the wrong thing).
the concept of archiving should be persistent throughout the tool, if it hides information in one place (ie the main initiative/objective page) it should hide it everywhere else (ie all dropdowns, related initiatives etc). feels like an oversight on the part of the product team not to think this through completely.
You really need to make this consistent!
If you are able to select an archived timeframe from the dropdown in a record, but archieved timeframes are NOT included in filters, for example on the workflow view. It can mean that a created item doesn't display in the main view. example creat and iniative - select an archived date. Then go to the workflow view of initiatives, and the date filter doesn't allow you to select archived dates, and archived dates are not included in the 'any timeframe' option so the new initiative has disappeared!