Agreed.A patient has a legal right to his or her medical records. To the best of my knowledge, even with a signed HIPAA, an insurance company has no such right.
While not quite the same, I always recommend the insured hound their doctor's office too when it comes to records. Its a lot harder to tell your patient to go pound sand, that you send the records whenever you feel like it.
I'll normally give the doctor's office (or service) a reasonable amount of time to respond (a few weeks) and then just cut and paste the Jetstream (or whoever) 4 or 5 contact attempts into an email and ask the client to make a call.
That normally gets the ball rolling.