why would they need to invade the Netherlands?
They don't. I've gamed this out many times & the Liege choke point can accommodate another German Army. The logistics trouble develops further south, beyond the Franco Belgian border. What difference that makes depends on how the campaign develops. Each itteration of this scenario had different decisions & outcomes in tactical and operational terms. In general I could not see a extra 4-5 corps of infantry and cavalry making a decisive difference in the west.
A further complication is the rapid Russian mobilization of the army group that attacked East Prussia. It was capable of overrunning the lightly defended territory to the Vistula River & investing the entrenched or fortified city of Thorn long before the campaign in the west is resolved. So, the usual result was the extra army in the west, and then some, packing off to the east to save Berlin ect...
Anyway, that my take from doing this one on the map or game board a dozen odd times.