I am 220 pounds and I ride pretty hard and do not find flex to be a problem with my steel Gunnar Roadie. This bike is built by Waterford Bikes in Waterford, WI. I had to wait a few weeks for delivery, but it was well worth the wait. I bought all the parts and built it myself, with some help from my brother since he is more mechanical inclined than I am.
My single speed bike is a Jamis Sputnik, which is also steel. The steel is not quite the quality of the Gunnar. I did have some flex from this bike, but I think much of that was from the wheels. The flex is mostly felt while standing during a steep switchback.
Since I started riding these steel bikes, along with a mostly aluminum Specialized S Works E5, I no longer own a plastic bike.