OP Amp controlling a Power Mosfet
The supply of the opamp is 0 and +5V from and independent supply

The minimum recommended supply rail for the NE5534 is 10 volts and its output will not get any closer to the negative rail than maybe about 2 volts so, given that the op-amp's negative rail is 0 volts in your circuit, the output will not get lower than about 2 volts and, the knock on effect of this is that it will always be turning on the MOSFET to a lesser or greater degree.
To compound the issue, the lowest input voltage that the NE5534 can adequately handle is only guaranteed to be 3 volts above the negative rail (or +3 volts in your circuit).

Surprisingly, an LM324 works exactly the way I expected: fine, but
this totally disoriented me!

The LM324 will work with supplies lower or higher than 5 volts and its output can be relied on to swing to within about 50 mV of the negative rail (0 volts in your case). It's inputs are good all the way to the negative rail too.

can an LM324 be used to control the gate of the mosfet? Or, is there
an other OPAmp that behaves as I need and is capable to drive the
mosfet?

It can be used but bear in mind that the input offset voltage of around 5 mV may give a battery discharge current error of a few percent.
Also, when driving a MOSFET like this, it's usual to turn the op-amp into an integrator so that the excessive capacitance of the MOSFET's gate-source does not cause the circuit to become unstable. However, of all the chips that are available, the LM324 is probably one of the least likely to be problematic in this respect. Having said that, I would still have a 1 kohm resistor from R1 to the -IN pin of the op-amp and, put a 10 nF local feedback capacitor from the op-amp output to -IN (just to be sure and sleep well at night).