H-bridge MOSFET NAND gate driver suffered from logic voltage drop
It isn't the input impedance of the CD40106B inverters that's the problem, it's the fact that you are totally overloading the outputs. Look at everything you have attached to each one on the MOSFET board:


Blue LED
two N-channel MOSFET gates
bipolar driver for two P-channel MOSFETs


You have 1k&Omega; pulldowns on the last two items in particular (3 loads), so it's no surprise that the inverter can't drive the control signals high.

The N-channel MOSFETs in particular need to have gate drivers, too, in order to supply the transient current that they need when switching.