Training center is basically just for leveling soldiers faster than combat.
Why not have the training center "produce" a random Lvl1 Soldier in that base every xx Days (10/15 days per soldier?)
They can come in only the basic P.P. classes, since "special" recruiting (Priest/Engineer/Infiltrator) are relevant to factions.
Also limit each base to one training center, since stacking these would fill up a base rather quickly.
Clarification - do not remove the EXP Training part from the Training Center, this would be an addition to the current usage.
Addition - Perhaps this functionality could come at the cost of an upgrade to the facility, instead of a native functionality.