Overflow drain for tub

I recently hired a plumber to install a replacement tub for our remodel. He installed the tub but didn't hook up the overflow. He said the previous tub didn't have an overflow hooked up and the the pipe sticking up was a vent pipe not for overflow. This is a manufactured home so I know some things are different but what is the point of an overflow if its not hooked up to a pipe but just dumps on the subfloor? My husband is of the opinion that the plumber was being lazy and didn't want to spend a few extra minutes doing the job right. What is the proper way to hook up the overflow in this case?

Crazy,was He really a plumber? Stop the check,that's outrageous!