Thank you, everyone, I am already learning a lot from your answers.
Put a 10-20ohms/5W resistor in parallel with the voltmeter's probes to identify the weak cells.
Would a 2W resistor be OK to use? If not, I will get the 5W one.
You should probably start from googling your device/battery manufacturer and model to see if anyone has done troubleshooting or rebuild. Simply replacing cells may not work at all if the process of battery recalibration or management circuit reset is unknown. The battery may lock itself when you disconnect a cell and you may have hard time to even get to the state you had.
Unfortunately I haven't found any project on the Internet dealing with this particular cell swap. But various vendors sell replacement cells for this pack, so that leads me to believe that there should not be any original cell lock or encryption in place.
First check what form the wiring from the battery to the device takes, just two conductors for power, or something with extra wires for data signalling?
In the former case, see if the device can be powered simply from a DC supply...
There are two extra wires, but they apper to be just from what I think is a thermal sensor that is placed between two cells.
Unfortunately I don't have any bench DC power supply. Maybe I should eventually get one, it would be useful for troubleshooting.
Just to be clear, I don't have the new cells yet. The behavior I descibed is happening with the old original cells.
I will get the cells, make a temporary pack from them (without permanent connections) and test if the motor is running without the original BMS. If yes, I will connect them to the orig. BMS (replace the original cells).
I will get the 18650 cells with welded tabs because I don't want to solder directly to the cells.
But soldering to the tabs should still be done quickly, right? I hope my 70W iron and lead-free solder will be good enough for that. I will practice on a completely non-functioning cell first. Is it a good idea to use some sort of heatsink on the battery terminal or a section of the tab?