Its very, very common for students to forget to pull the gear
up on missed, I"m sure adding complexity with flaps would make that
worse.

For this reason I think it's sometimes a good idea to teach a student to use
flaps on approach. If the training is in a fixed-gear but the pilot plans to
fly a retractable later on, using flaps from the beginning helps establish the
habit of reconfiguring the airplane at (or near) the FAF and again on the
missed.