^That's not really a contradiction. They didn't give him details of what they actually did to River, just what the results were.
Simon did not break River out on his own. There's no way he could have. He needed to acquire the uniform he was wearing, the various security clearances needed to get in, someone to fly/drive the getaway vehicle, someone to get the plans of the facility and so on and so on.
It makes sense for him to go in because, not knowing what state River might be in, she may have needed immediate medical attention before being extracted, and him being a familiar face doesn't hurt either.