I picked up a screw too a few months back. Luckily the leak was slow enough to top up the air and ride it the 4 miles to my local bike shop. The guys I use are proper pros and told me they would plug the hole and patch from the inside in all cases bar a well worn tire. I told him I was about to go on tour for over 1000 miles and he said it would be perfectly safe, even ridden hard. I can tell you I was relieved as I was not willing to scrap a 2 month old Michelin.
My opinion would be to trust in what the tire bay tells you. Trustworthy ones don't drum up business on the sly, good places are always busy naturally. My guys gave the bike a once over, despite having passed its MOT only months earlier and told me straight - the chain has seen better days but not urgent, the front brakes have only about a 1000 miles left in them and the rears are half worn. Otherwise, go enjoy.
They were absolutely right. I toured, got back and changed the pads. Just a few hundred miles left on them but still pulling up safe and true.
The guy also showed me how to tell when a chain is knackered. You can learn a lot by keepin your ears on at these palces IMHO.