There's nothing dangerous about a few minutes of cold, since a 1 year old can regulate body temperature that's not how kids get sick, but a miserably overheated child in a car is no fun.
You're right that safety in the car is primary, so if you have a zippered attachment designed for that purpose, great. Otherwise you can often tuck a lightweight blanket (particularly the kind that traps heat in the material, like an afghan, can be tucked into outer straps or even behind the seat with a clothespin.
For going into a store, I would just keep him bundled in the blanket until I got inside (whether carrying or using a stroller), or even plopping him in a shopping cart that's outside in the parking lot, and not worry about the coat. For the stroller, you can also put the coat on backwards - put the arms in (keep his hands inside the cuffs, not out in the cold, and you won't need mittens). Then just lay the rest of the coat on top like a little blanket, no zipping in the back. It also works if you put the coat on that way and hold his back against your chest while you walk inside to get a shopping cart. A hat is plenty for the 60 seconds it takes you to get inside.
Coats are more important for sustained time outdoors, like in a stroller or on a kiddie sled. Then you need the full regalia because they aren't active at all (so they need more than you do because you are walking and they are not).
