I honestly thought that all the comments on this thread were about children sitting in the designated seat, and thought everyone had become neurotic all of a sudden. It never occurred to me that people were talking about them sitting in the body of the trolley. This is yet another reason why online shopping is so good.
I haven't done regular supermarket shopping since my two were little, and I put one in the toddler seat and the other walked by my side, or was sent to find items that he could easily spot and reach and were no more than a few feet away, to stop him from getting bored. It was tedious, and as I worked full time we had to go on my only useful day off, a Saturday, when the place was full of other families in the same boat. I don't drive, so my husband had to come too - fun for all the family!
I don't think it's a great idea to put toddlers in the body of the trolley for safety reasons, but I doubt there is a serious hygiene risk compared to sitting on a train or bus, or a lot of the other day to day things we do as a matter of course.