Here are your parts
Parts for Whirlpool LER3624EQ1 Dryer - AppliancePartsPros.com

See the attachment for the tech sheet.

The operators manual and the wiring diagram cover a number of units so I am not 100% sure why you do not have a wire at V.
If wire at V is just for an operating thermostat that has an internal heater which is used for temperature selection so if your unit does not have temperature selection you will not have a wire at V.

Does it do this in both timed dry and auto modes?
This is important because the timer motor gets power from different circuit paths through Timer switch 0 depending on the mode.

In timed dry the path is direct from Neutral through TM/WB contacts,through the timer motor and then to L1.

In auto modes the timer motor only gets power when the heater is off.
L2 voltage is then at the left side of the heating element, the resistor reduces it to 120 volts then through TM/OR timer contacts then the timer motor and finally L1.
Note that when the heater is on L1 voltage is on the left side of the heater so the circuit ends up being L1 to L1 and no current flows through the timer motor.

Perhaps te timer chart will help you figure this one out as I cannot say what is causing the problem.