Tires with more mileage are more susceptible to flats. They are thinner and (depending on how well one checks and cleans tires) there may more be residual material still emedded in the rubber that can work its way thru. To add some pure speculation the older rubber may also be less resistant to debris.
I've ridden centuries with rear tire threads showing, but if going for a long ride I would prefer to start wtih a fresher tire so as to reduce the chance of a flat.
Some tires (notably GP4000S) I've used until threads show (and usually thin flat bits of rubber start to peel away) with no flats the entire ~4.5k mi life of tire.