What could cause my washing machine to stop before the rinse cycle?
Sounds like the timer-switch is dirty or worn, based on the fact there's a sweet spot that works. If the timer is stuck, i.e. it never reaches the end of the cycle on it's own, the gear teeth are probably worn in that spot.

If the timer works, it just 'misses' the rinse, you may be able to access the contacts and clean them. If the gears are worn or broken, you'll probably have to replace the whole assembly.

Since you can get it to work in the sweet spot, that rules out the usual suspects like control valves, relays, pumps.