The only dryer I found that has a 3770 ohm resister is the GE/Hotpoint older design type using the
diagram found here
It looks like it is part of the Auto normal and auto knits cycle, normally the timer motor stops until the dryer reaches set temperature then when the heating element kicks off the timer is allowed to advance when the timer is selected for low heat or Polly knits this resistor is by passed and the timer motor keeps running even when the Polly knit thermostat shuts off (low temp)
The confusion was that you stated that the resistor is parallel with the start button, other then maybe the start button has a couple extra un connected terminals that was used as a connivance to locate the resistor I found no diagrams showing any resistors in the start button circuit, almost all start buttons apply power to the motor start windings until the run relay pulls in, this same relay also provides L-2 power to the other side of the heating elements. most other brands use a very similar circuit but most use a 4.7k ohm resistor or close to this, GE/Hot Point is the only one I find with the 3.77k ohm resistor?.
As far as what would this bad resistor cause if left bad? timer motor would not advance in any of the auto settings but would have worked just find in any of the manual timed or low temperature settings
If this dryer is not a GE/Hot Point let me know which one and maybe I can pull up a wiring diagram.