I'm trying to figure out why a handheld vacuum doesn't turn on (I have a red light). I tested the battery connected to the DC motor, and it works perfectly.
So, I'm wondering if when everything is connected to the circuit board, it may not allow the DC motor to turn on because of an unbalanced battery, and it turns on the red light. Does it make sense?

 

It's not likely that a "throw-away" consumer product would have that level of sophistication.
( it's not likely that the appliance has individual Cell monitoring in the Battery )

But, it is possible, ( especially with the cheapest possible Battery available ),
that one of the Cells could have gone dead.
This will prevent proper Charging,
and possibly create an apparent High-Impedance "open-Circuit" in the Battery,
this may result in a "Low-Voltage-Shutdown" in an attempt to prevent damage to the Battery.

I would dispose of the Vacuum as soon as possible, it is potentially a FIRE-HAZARD.
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Disconnect one lead to the motor, turn ON the switch and measure the voltage. Post it here.
Temporarily wire a 1kΩ resistor across the battery leads (while still disconnected) and measure the voltage across the resistor. Post it here. Third, if you have a standalone power supply, set it to the nominal battery voltage and temporarily connect it to the motor leads. Watch polarity. Monitor current draw if your power supply allows you to do so. Only run the motor briefly in this test. It’s risky and could burn the motor up.

These tests will show if the batteries are good and while they might be able to supply enough voltage - they can’t supply enough current.