Yes, do not have the owner do the vet exam. Sometimes the vet doesn't note things because he knows and likes the owner. I had this happen when selling a mare. The buyer wanted a vet check, but wanted me to call my vet and pay for it. I did, but the vet just showed up, looked at the horse briefly and wrote out a paper saying she was healthy and sound. I didn't really know what to ask him to do at the time, and the buyer didn't specify. He was trying to help me out I guess? I let it slide because I was not happy about having to pay for it with no guarantee the lady would buy, and because I was high school age and didn't really know what to do.
PS: This mare was sound as far as I knew. He didn't flex her though.