I have built systems on carpet in socks before and not used the strap, it just gets in the way. I usually just grab the case, with PSU unplugged, and go about whatever I need to go. I have never lost a component before, though it doesn't mean I may not in the future but for now I have not.
If its unplugged that means you aren't grounded at all, even by touching the case.
That's
really bad practice.
All mine, but even if they were not I still wouldn't. All they offer is to get in my way.
The point isn't whether or not touching the case is enough, the point is that if you're working on someone's computer (even if they aren't paying for it), then you should be professional and make sure you take all the precautions that you can.
Static damage isn't something that you can just go "oh, thats static damage", so if you do irreparable damage to someones computer then there isn't any way for them to trace it back to you and get the fair compensation they deserve. Which for me at least is a reason to be as cautious as possible.