Okay this is somewhat of a mystery since first of all it works through the wash cycle which suggests that there is nothing mechanically wrong with the machine, and second, all that has to happen is for the inlet valve to open to let cold water in to rinse.

If you are using only hot water for wash, try a wash cycle on the cold water and see if it fills. If it does not and you get the smoke and burning smell the problem could be in the inlet valve assembly or the timer, but most likely the valve assembly. To test, hook up the cold water connectors at the valve assembly to the hot water and see if it fills on cold water wash. If so that wold confirm the valve assembly as the problem. If not, the timer is likely the issue.