People, including some on here, with unspent and far more serious convictions, including jail time and sexual offences, have got to the US on an ESTA with no problem. I would be absolutely staggered if a 10-year-old caution flagged up with them. Also, you don't have to declare all convictions, only ones relating to moral turpitude (now reworded).
According to our beloved Home Office, the US has no routine access to the PNC (https://www.gov.uk/government/publications/access-by-the-us-authorities-to-information-held-on-the-police-national-computer). They can only access your info by doing a specific individual request to the UK authorities if they have grounds for suspecting you of something.
A dummy run would of course do no harm and would make you feel less nervous when it comes to the family trip. If you have any choice in the matter I would recommend going through Ireland, as they have US border control on Irish soil at both Dublin and Shannon, so in the incredibly unlikely event you are refused entry, you only have to worry about returning from Ireland and not from the US on the next flight at your own expense. Though of course with work you may not have a choice in the route, but like I said, people with far worse have gone there no problem.