This actually formed part of my experience with UPS. When the guy said to me "How much does this TV weigh?" I had no idea: the TV was in my parents' house a few hundred miles away. And why the fuck should I have to weigh it? He wanted my parents to take a heavy TV set and put it on some scales in order to weigh it. When I told him to make it up he said "but we might get it wrong and undercharge you!" I don't care. Just charge me a penalty or something. Shipping is one of those things that I don't have any sort of inclination as to how much a competitive price is.
Amazon uses something similar to this if you order a TV from them - they have someone who actually brings the TV into your house, unwraps it, connects it and takes away the box.
Again, ad nauseum, I am not in any way suggesting that UPS or FedEx should be doing this.
I know you're frustrated by folks saying that you're suggesting UPS or Fedex do this, but your post title suggests that's what you're asking for, and it's also rational to view it as more likely that those carriers would create this service, or someone would create it as an overlay (MVNO-style) utilizing their existing shipping networks, than creating an entire new shipping system - the cost would just be astronomical to create a whole new transport network for this.
Don't have a scale to weigh a package that feels about the same as a pound of butter? Enter the weight as 5 lbs. They'll happily charge you more and deliver it.