No, I don't think he feels this way.
My dd has an active social life. She has lots of friends and is genuinely happy. She has no behavioral problems and makes friends easily. There is really no basis for him to make the comment on her social life.
As for constantly testing. My dd got tested (IQ) for entrance to the gifted school when she was 5 1/2 and then again for the Davidson Program (IQ and achievement - Previous IQ was too old and they needed a more updated one) when she was 7 1/2. She has taken the SCAT to qualify for summer programs thru CTY once. These are all the tests she has taken in 8 years. He school doesn't even do end of year testing (is a private school).
He really is unhappy about the grade skip and made it clear to the school that he did not agree. Fortunately I can legally approve it.