I've never been on a velodrome but I do ride a fixed gear bike. I love the way it makes me feel connected to the bike. Your feet have to do a revolution almost as frequently as your wheels. You never space out to the experience the way you do on a coasting bike.
Occasionally I wish I had a different kind of bike but that is very rare. Riding a fixie makes me want to ride more often. Its the way bikes were made to be ridden