I think there's definitely a cultural element to this. I'm in New England where there's a lot of perceived virtue in the old adage "Use it up, wear it out, make it do, or do without." And there's also an element of "old money" here so people wearing all shiny new things, driving new cars, etc are looked at by some as "nouveau riche" (very different from Beijing, I suspect!).
I have a hard time being bothered by visible wear on most things. I probably wouldn't wear that coat to a job interview or important meeting, but for everyday wear I would live with it until I found a great replacement. I would also remind myself that most of the time, the bag itself will be covering the wear, and when I take off the bag, I'm probably taking off the coat as well, so how much will people really notice it?
Like Jenni NZ, my frugal side would be bothered by coats that wore out quickly, especially the pricier one. So when shopping for a replacement, I would be looking for a solution to the problem, maybe a smooth, durable fabric that could tolerate a lot of abrasion. A trench coat with a warm lining, perhaps?