Hotend heating failure too short before limit
There are many unknowns at this point, you need to work methodically through each stage. You probably also need some test equipment, at least a basic multimeter. Ideally a temperature probe too.

First, check that your power supply can drive the hot-end directly. You already know how long it takes to reach 120°C, so be sure to not let it heat too far beyond this time. This will bypass all of the control logic, and risks overheating your hotend/burning out the heater - but it should be OK if you limit to a few minutes.

Check the temperature reading, and cross check with some PLA or other low melting point filament to see if you actually reach ~180°C. Check the voltage as close to the heating cartridge as you have a connection point. It should be at least 11V (with a 12V PSU). My guess is that you have a bad connection somewhere...

If you observe the hotend getting hot enough to melt plastic, check the thermistor readings. You should see a fairly constant rate of heating well past 200°C. However, I expect this will work fine.

Finally, check the hotend mosfet drive and output as the hotend reaches the target temperature. It should be on, then switching on/off, then off as you pass the target.

If this all looks OK, repeat but with the board in-circuit and controlling the heater. It may be that the mosfet is damaged, or it is not being driven very well - but to progress past that point we need to see voltage readings at various points.