Not only do we still have baggers and cashiers, but some Kroger stores have a service called Clicklist where the customer orders groceries online, then drives to the store so a staff member can bring the order out to the parking lot and load the car.
To a European the whole idea of having baggers sounds completely idiotic. People that get paid to put your groceries in plastic bags? I don't think we've ever had that. But grocery pick-up service is widely used where I live. Because parking is a huge problem, it's much more convenient to order them online and pick them up. As I don't have a car at all, I have my groceries delivered when I can find a coupon for free delivery. I order a couple months worth of staples and they carry them into the kitchen!
Baggers often do a lot more than just bag groceries. They sweep floor, take items to a customer's car or man the drop off desk, collect the shopping carts, take out trash, pickup the parking lot, change out mats, conduct price checks, collect a customer forgotten item, often refill the supplies at the cashier stands, wrap flowers, etc. At many large stores, there will not be one bagger for every lane, but they will help out the heavy orders so the cashier is not lifting / pushing / bagging a lot of items and slowing down. If you think about it, your stores likely have someone doing all the above, except bagging task specifically.
The big difference, I think, is the frequency in shopping. In Canada, with a family, I would have one large trip per month, and several small trips for fresh items (milk, fruit, bread). The large trip may be $300 in groceries. I have even had two carts at times.
In europe, three very large bags is about my limit, although I never took a car to the store in europe, and the kitchen was small too.