I've been thinking about this, and the chances of having dog shit brought into the house on someone's feet. I agree that it would/is horrible, but I really don't think it's very likely.
For a couple of years I worked in a community centre which was mostly carpeted. At the front doors there was the type of fitted, heavy duty doormat you often see in public buildings.
Over the course of a week, there were at least 1,000 visitors to the Centre. Some of those visitors would go out at some point during the day, and come back in again, so for the purposes of carpet dirtying, they could be counted as two visitors. We were open 7 days a week, and only really closed fully for less than a week in total, but to account for quiet time, I'll assume a year of 45 weeks. 45,000 visitors.
I worked there for 2 years - so, 90,000 visitors in that time. None of those 90,000 visitors took their shoes off when they came into the Centre. None of those 90,000 visitors over the course of the 2 years I worked there (I was the manager so would have been told if this had happened) brought dog shit into the Centre on their shoes. Not once!
The carpet was stained in places, and I didn't think it was the best choice of flooring for a busy public building, but none of those stains were caused by dirt or shit or anything brought in on people's shoes. They were mostly caused by people dropping food or spilling drinks on the carpet.
90,000 visitors - all with shoes on - no dog shit or anything else that would wreck a carpet.