I've been riding an 80-year-old Columbia recently. Single-speed, coaster brake, about as basic a bike as you get. A 35 lb. "Lightweight." When you consider the mass of the rider/bicycle as a system, having a modern lightweight bike doesn't make that much difference. I know there are a lot of Spandex-clad folks dressed up like they're running the Tour de France on 5 lb bikes made of rags 'n glue that say otherwise, but physics is physics. They go faster because they're stronger and they weigh less than me, not because their bikes are lighter than mine. I've read this whole 2-year thread. Lots of talk of gear ratios on single-speeds, multi-speed bikes, etc. But nobody has mentioned what people used to do back in the day when confronted with a hill: Dismount and walk your bike uphill. You get to see your bike from another perspective and life slows down for a while. You still get all the exercise.
My mom and dad took the opportunity to bike the newly completed Blue Ridge Parkway/Skyline Drive during WWII because there was gasoline rationing and no automobile traffic . My mom was riding a 3-speed, 60 lb. Raleigh and dad was riding the same 80-year-old Columbia I ride today. He said next time, he'd leave the bikes at home and do the same trip with shopping carts.