When you replaced the start button, did you replace the switch behind it as well? If the contacts in the switch are not touching, the dryer won't start.
To get to the switch and see if it is causing the problem, follow the step-by-step instructions found here at FixYa.com:
http://www.fixya.com/support/r6019584-access_start_switch_kenmore_70_series
Greg
October 2010