The last one I picked up off the street had no problems whatsoever, except for the fact that the power cord had been cut off. Presumably someone had cut it off deliberately, which is infuriating - but hey, free toaster.
People usually do that to specifically indicate that it has a problem of some sort, so someone else doesn't assume it's working. Although with a toaster, just the act of flipping it around to cut the cord might dislodge whatever toast bits were causing it to smoke and such in the first place.
In my experience, desperate people cut the cords off of perfectly good appliances to sell the copper as scrap. Copper wire fetches something like ~$1/lb as scrap. It's infuriating when they cut it right at the connection to the appliance, making it too time consuming to test.
A label of "Broken" or "Works" or "Missing X" is so much more helpful for appliances left out.
Good point, if it was just sitting by the curb, a metal scraper might have just cut the cord instead of taking the whole thing. I've noticed that different scrapers have different criteria for what they'll take. Some will take the whole thing and break it down or just sell it as unsorted metal and others are more particular and will pick through and just take what they know they can get the most/easiest money for.
I once had to cut the power cord off of an appliance (a dehumidifier, in this case) as part of the process to claim a recall. They company didn't want me to ship back the entire unit (annoying, but understandable given the weight of it), but they wanted to ensure that it wouldn't still be used, since it was recalled for being a fire risk.