I hear fixed gear riders talk all the time about how they are so connected with the road. They talk about having ultimate control over the speed of their rear wheel like it matters. I think it's especially silly when they say that kind of stuff without being clipped in.
I feel connected to the road when I tune in to that feeling. When I pay attention to the feedback that I get from the bike. When I feel the uneven pavement beneath my tires. It doesn't matter if I have a more complex drivetrain on my bike; I feel the same. Don't get me wrong, each bike provides different kinds of feedback (for instance, the mountain bikes have a front suspension). But beyond mechanical efficiency, there is no inherent superiority in a fixed drive system.
I've been riding a fixie for over a year now, so it's not that I'm so new to the bike that I can't connect. I have 4 bikes now and each of them feels different. I can appreciate those differences and hopefully use the bike that best suits my needs. The Diamondback 26" is great for getting around, hauling small stuff, having fun, and light off-roading. The Kona 29er is great for off-road and kind of a bear to push on pavement. The Trek 1200 is my choice for long rides. The KHS fixie is a great commuter. Each bike works different leg muscles. Each bike responds differently to flying down a hill. Each bike has its own unique personality (although I'm still getting acquainted with the Kona).
If you want to feel connected to your bike, just ride it. Ride it a lot. Everywhere. Every day. You can connect with your cruiser, recumbent, hybrid, TT, road, mountain, fixed, single speed, or vintage bike. You can get high on cycling while retaining the ability to coast. I encourage you to do so.