I want to identify current tasks -- those that have start dates on or before a particular date (usually "today"), and that are not yet checked DONE or moved to a "done" list.
You can search issue other than <today> by using the operator "-" (without the quote) and combine them with open cards eg.
-created:day is:open -list:"done"
However, there are no operator for <before> today unfortunately but you can specify how many days before today or use the operator "-" for the opposite.
Feel free to have a read of this documentation: Searching for Cards (All Boards)
Hope this helps.
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