I use this site to lay out foot-race courses for our local community celebration each summer. We have a fixed finish line for our courses so I would like to be able to shift the start line to zero in on an exact distance. I believe that mapmyrun.com has this feature.
Great idea, and good news, the site has it! I guess your question means that it's a bit hidden so I'll work on improving that.
Anyway, click anywhere on the blue line when you've created your course, and you'll see a sequence of dots appear along the line. The big dots are points you set in the map, the little ones are points the routing algorithms automatically added. There should be a big dot at your starting line -- just drag that to the spot you want to move the starting line to! Note that the line doesn't refresh until after you place the big dot.
Give it a try and let me know if this works for you!