The hardest thing to find when it comes to vehicle repairs is (in my opinion) understanding.
Quite often the people on both sides of the counter are not on the same page.
I tend to think there is some lack of understanding going on with this particular problem.
Apparently the OP did not know what a radiator was ( not meant as an insult ). Not everyone has someone to explain things to them. The repair shop could be excused for not realizing that. But the OP should have asked more questions when paying for the repairs. Without seeing the invoice we can only assume the OP got what they paid for.
“Without seeing the invoice we can only assume the OP got what they paid for.”
I Agree. It Seems That The Original Complaint About Over-Heating Was Resolved.
There Must Have Been Estimates Given And Approved, Eh?
In my state an estimate must be given and a repair cannot exceed a customer approved estimate by 10%.
CSA
To the OP, the radiators in old cars (circa '70’s) were easy to see. Now the motor compartment is covered in plastic, there are shrouds to direct air flow, and there are oil coolers, and AC condensers sometimes mounted in front, and beside the actual radiator.
Since your SUV is rather “tall” you might need a step stool and a flashlight to get high enough and lean over far enough into the engine bay to look back to the front to see the new radiator. If you do you should see fins and tubing that look new, have no bugs stuck in them, and that can confirm you got the new part you paid for.
In NY, NJ and PA where I’ve lived I believe there is a law that says the auto repair shop has to save the old parts for the owner to examine when you pick up the car post repair. I ask my mechanic to save the parts because I like to see just what caused the problem and see if I can learn something. Next time you take your car in for a repair ask to see the old parts the shop replaced - I’m sure the shop will comply and you will feel better about what was done and why. If a shop refused my request to see the old parts I’d take my business elsewhere.
@UncleTurbo
The folks at The Department of Redundancy Department thank you.
CSA 
I try to read the other responses first but sometimes they are just too cumbersome and many so I don’t. But then I usually say that I have not read the other responses.