WTF is this one: "What if I need to pee mid-run"? You just stop, optionally pause your watch, and pee. If you are doing any sort of tempo or intervals, you should hold it for a moment so that it doesn't mess with the purpose of your workout.
Now pooping mid-run, that might be a problem depending on where you run. When at home, I always take a few pieces of TP with me when I go running and occasionally I need to use it in not-so-appropriate places. When I was in a training camp in Kenya, that was worse. No matter where you run, there will always be people around, and every spot of land is someone's corn or passionfruit patch. I had to wake up real early so that I definitely get that bowel movement before setting out.
As for excuses, when my coach writes me a rest day, I have to really try hard not to run at least a 40 minute easy jog, so I have the opposite problem. When I'm told to rest and it's icy outside, it helps me to actually rest!