501 framesets would not typically get you into weight weenie zone, compared to more lithe tubesets like the 753 or Supervitus 980, but would usually build up to bikes that weigh around the plus/minus 23 pound region, using mid level components, which is really not bad at all, considering the robust nature of 501 frames.
Something like a Brooks saddle would not help in the weight control effort as much as a new nylon or CF based saddle, so you can make a special effort to carefully consider your compnent choices to minimize the weight gains as you build up the bike. and possbily come up with a bike even lighter than most expect with a 501 Reynolds frame.Anyway, I don't think you can consider a classic steel racing/sport bike as "heavy" as long as it does not get too close to 25 pounds and up.
JMOs

Chombi