Thanks to Jonathan Drummey, I was able to resolve this. When I said you MUST have a Date field for a line chart to work, I was referring to the "Show Me" option which suggests charts...
Jonathan's response sums up how to resolve this problem, should anyone ever run into a similar scenario in the future:
"Show Me will require certain pill types/data types, and when you have the Mark Type set to Automatic Tableau will make choices based on the pill types/data types. However, you can always manually set your own Mark Type for most anything, though not all pill types/data types make sense. So, line charts *don't* require a date field.
You can have a line chart with continuous and/or discrete pills on Rows and Columns. There are some restrictions about connecting line sections when you have more than one discrete pill on the same shelf (Rows or Columns), but that's about it."