TB6612FNG DC Motor Driver Low Current Problem
I expect “peak current” from driver is limited by either power dissipation in the driver and/or the duration of ESR*Cout used by the driver and/or the current limit in your supply or output impedance.  

This may be shorter than the L/DCR current rise time of the motor, which is further slowed down by mechanical load.  

Your shared supply current to board must be able to supply 2 stall currents at rated voltage to ensure stable operation.

For at least one or more of these reasons, your motor is not getting the rated current due to a drop in driver voltage.

Driver Ron = 0.5Ω @ 5V. (Typ) not worst case. @25’C

Motor DCR= 6V/2.9A=2.07 Ω using stall current.

supply ESR= TBD or active current limit = TBD

With more details on your measurements at all nodes, a solution can be suggested. like a heatsink for IC to prevent rise in Ron and thermal shutdown.