I would have the tooth extraced by a dentist that removes the periodontal ligament and cleans out the hole that is left where the tooth was at. That is to avoid a cavitation later on.
Of course your dentist messed up. You are correct. You will never get any dentist to admit that they messed up so I would suggest not even wasting your time and money there.
If you go back there prepare your mind before you go. Do not let them talk you into having a root canal done under any circumstances. They will talk and talk and try and pressure you into it also. ALL root canals fail. They are traps for bacteria.