Criminal background check is not so reliable as it's made out to be.
Someone who got a criminal background check to show parents could just as easily commit a crime the very next day!
Therefore, you have a "reassured" parent on the basis of that criminal background check, unaware that the caregiver is currently facing criminal charges. It gives false reassurance.
Parents shouldn't just go through the motion of doing interviews asking all the "right" questions from a questionnaire they got online. Well, not only parents looking for daycares read those questionaires! How many daycare operators read them and prepare themselves to give the "right" answers? Answers that they know parents want to hear?
A lot should also be based on keen observation/assessment...and that gut intuition. Ideally, a parent should come during daycare hours and observe the interaction between children and caregiver.