Just have gone down similar woes....see my other posts.

First catching the fluid is no excuse for having Bozo the joke clown hammer on your car on your nickel. I'm sure the cooling system was in sucko condition...but Spiffo Bloob, for a coolant flush? Dude, you need to hang here more often. Not lecturing, just logic.

Yes coolant should be treated as the toxic chemical it is. I splashed some on my lip and got a blister! First time in 30 years of car wrenching this has happened to me. But it's not a chlorinated solvent, not PCB, etc. It does readily break down....and yes in some areas it's OK down the drain.

OK as for your system, yes a new radiator, hoses, thermostat (and seal), maybe a new goose-neck, and lastly:

The little vent hose from the rad nipple to the coolant expansion tank. If clogged (with gunk) you will have all sorts of problems. It clogs pretty easily. I'll bet that hack place didn't even blow the hose out...if they even did a flush.
--
Paul's Amsoil and other lubricants