There are several things that determine whether or not a plane will self-recover from a "hands-off" dive & the flat bottomed wing is only one of them. My Superstar hot rods will dive forever, flat-bottomed wings notwithstanding. Trim, C-of-G location, incidence, relative positions of AC to C-of-G, & relative positions of center-of-drag & cente-of-lift are far more powerfull factors than the airfoil configuration.

Getting back to the second plane question -- standard trainers can be easily converted to highly capable aerobats, & hence, make excellent second planes. They can combine relaxed (relatively) flying capability with powerfull aerobatic capability in a package that looks deceptively like a trainer (until you pull its tail) & costs little to engineer.