If the whole point is to find discrepancies between what the prospective employee tells you and what is reality, not highlighting differences between those two sets of data is an issue, even if because the people doing the background check just lack access.

The real reason most background checks happen are so that companies have someone to point the finger at when one of their employees breaks the law. If that wasn't the case then you wouldn't get into situations like background check companies reporting completely clear records when the government has a simple process to show you every sentence given to someone. They stick to third party companies because they are cheaper and don't trigger compliance regulations required by law.

They would go for the better, if slightly more expensive, data source in a heart beat if finding about criminal history was the actual goal