An extreme approximation I would make is at maximum 100 amps going through the mosfet for a few seconds when the motor is heavily loaded. I think I will go with two paralleled mosfets just to be on the safe side.

One last question : would you parallel two CSD19506KCS to spread the heat or one is enough in this case ? Thank you again for your time and knowledge

Ok, I think I know what to do next to improve my circuit : layout the control circuit using manhattan construction, use a ~1-2 amps gate driver, very thick and short driver-gate connection, use deadbug construction for the power switching circuit with ground plane and very short and thick leads , use 1000 µF cap, replace IRFP2907 with CSD19506KCS, screw mosfet to a cpu heatsink, use twisted pairs to connect the motor and the batteries to the power switching circuit

Thanks a lot for your edit, I think it allowed me to finally understand what you meant. Please take a look at the edit in my question : I've made a schematic drawing equivalent of your PCB; please tell me if I'm wrong.

Sorry, I think I wasn't clear enough in my comment. I meant to say that the CSD19506KCS is superior in every way to the IRFP2907. Therefore there is no benefit in using an IRFP2907. Performance of the CSD19506KCS is similar but its gate charge is much smaller, which means less drive power required, faster switching time and therefore much better efficiency with less thermal losses.

Deadbug construction is soldering component leads to make nodes that are suspended mid-air, whereas Manhattan is soldering these nodes on islands or pads that are right next to a copper ground plane, is that right ? Manhattan seems much easier to make and clearer visually that perfboard. How do you accomodate DIP through hole IC packages on a Manhattan construction though ? Do you extend the pins with wires and suspend the IC or you make smaller pads ?

"Just make sure the switching happens all in one place, keep the MOSFET-diode-bypass loop short, and you'll be sitting well." Could you please detail this for me ? Especially the mosfet-diode-capacitor setup, I haven't understood it properly and I feel this is a key element in designing power mosfet circuits.

Looking at the CSD19506KCS datasheet, there is absolutely no benefit in using a IRFP2907 over it ? (I bought a bunch of them as well) Where can I get information regarding the latest parts and how can I know if a part is outdated ?

If I've understood correctly, by putting a diode + capacitor as close as possible across the mosfet drain and source, we deal with all the switching noise and transients in one area to prevent it from interacting with the rest of the circuit ?

Regarding pairing conductors, how can all the inductances be in series when there is a battery in the middle ? You suggest shortening the wire between the motor and battery as much as possible so that the two wires going to the drive can be twisted together to reduce coupling ?

After a few reads of your answer, here are some questions I have regarding what you told me : I've read everywhere that putting a flyback diode + a snubber across a motor is always a good idea; yet you say that we can move the diode across the mosfet. Will this be equivalent ? Shouldn't we instead add a diode across the mosfet ? Isn't the body diode supposed to play that role for the mosfet ? (maybe its performances are poor compared to a proper diode)

Thank you very much for your great answer and the time you took to write it ! I'm indeed an electronics newbie just starting to understand practical circuit design and all the specific advice and good practices you gave me are what I'm looking for. It will take me a few days to process everything you wrote so I might come up with more questions later on.

Just edited my question to add the parts list I used

MOSFET is an IRFP2907, I chose it because it is rated for high continuous current and has a low Rds on. What do you mean by old and slow MOSFET ? What would you replace it by ?

@Tim Williams - I've added a picture showing the entire layout : motor and main batteries are connected using car battery wire (cables will be shorter in final application). The MOSFET is under the heatsink, MOSFET drain is connected to the heatsink and upper aluminium bar. The black box is a switch. MOSFET source pin is pressed against the lower aluminium bar by a washer visible at the bottom. Both grounds are connected by the black source wire. Capacitor across the batteries is a non-polarised electrolytic motor starting cap.