Copenhagenize advocates normal people riding bikes, and those who already ride setting an example to non-riders that you don't have to be a fitness nut to ride to work and other destinations.
Sure Danes sweat, but sweat is less of a social taboo in countries where active transportation is mainstream. They also don't ride particularly fast or far, so the sweat isn't drenching, just a bit of dampness.
I live in a sprawling Canadian city, but by simply choosing my home and my employment to both be in the city centre, I have a surprisingly similar lifestyle. Work is about 2km from home, so yeah, I just wear whatever I'm planning to wear for the day. Typically, I have to mop off my forehead when I arrive, but that's it. And anyone could make the same lifestyle choices without having to be unusually fit or invest in expensive equipment.
If you bike a significant distance (>5km or so), or have hills to negotiate, or insist on riding fast, then your needs become different. But biking like you do will never be mainstream, it will always be an exercise regimen to most people rather than an efficient way of getting to their destination.