My Whirlpool LSB6400KQ1 runs through the wash cycle but stops at the rinse cycle. No other cycles rinse either. Lid switch seems to be working as I hear a "click" when I close the lid.

Hi from retired Englishman in SW France,
(worth trying this but no promises!)
I would gently suggest as a first step that you clean the drain motor lint tray/filter. So many 'faults' are caused by blocked filters and many machines will not even go to spin or release the door lock if there is any water still left inside.
Here are my complete guidance notes for checking both the drain motor filter and the drain motor ;-0)
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If you are unsure where the filter is, it is often behind the front lower (kick) panel (or in some machines at the rear in an awkward position!) and most are held in by a round cover/knob. Sometimes the filter will not come out with the cover and you will have to persuade it to come out to play!

REMOVE POWER AND ENSURE YOU HAVE A WATER RECEPTACLE/RAGS TO SOAK UP THE WATER WHICH IS BOUND TO POUR OUT!
It's surprising what people have found in theirs! It is always worthwhile checking it on a regular basis.
Now- whilst the filter is removed it is normally possible to see the drain motor and ensure it is not jammed by a foreign object such as a needle or a toothpick. It should be able to be turned by hand/finger- the power IS OFF!
At this point it is also possible to test if the motor works. No clothes or water in the machine, select a spin cycle and put the power back on. You should be able to see the motor turning at some point, possibly straight away. Using something suitable like a glove see if it is easy to briefly slow the motor down. If it is easy then the motor is malfunctioning!

Good luck!
John C
If this is the solution please score accordingly- thanks!