Great answer. Just to add, this is with red meat where E. Coli is the main concern. That exists in the gut of the animal and that bacteria can travel to the surface of the muscles during the butchering process. Poultry on the other hand is mainly concerned with Salmonella which does live in the muscles of the animal. That's why people aren't ordering rare chicken breast.
Also, rare chicken breast is apparently pretty unpleasant to eat. Kenji Lopez-Alt did an article where he discussed how you can use sous vide to pasteurize chicken while still keeping it at a low temp, but the result (while perfectly safe to eat) was not enjoyable.
It was actually one of the first things I tried when I got my sous vide. Amazingly tender and juicy but the texture reminded me too much of under cooked chicken that it made me queasy. Now I stick with steak sous vide.
Somewhere between the stomach and the anus fecal matter is created. E. Coli lives along the entire tract. You're not wrong, but it's not just fecal matter which contains E. Coli. Basically the entire process involves it. Stating that shit on your meat is the source of E. Coli is understating the issue but it also triggers a more visceral reaction.
The stuff which causes you to have diarrhea that gets on the news is caused by the contents of the digestive tract being spread over your meat. Whether it's technically fecal matter or not doesn't really matter if it's stuff that will cause YOU to lose control of our bowels.
Proper butchering SHOULDN'T spread the contents of the digestive tract on the large muscles.
