Exactly what I did when the motor packed up on mine. Ditched the motor & fan, wired the hood switch to the new remote fan, kept the switch & power to the hood lights.
New inline fan off eBay iirc 40 quid, mounted mid way along the duct in my case, very quiet as in next store room to hood, very good extract rate.
Only thing I couldn't be arsed working out was how the 3 speed controller on the hood worked - to see if I could use that with the new fan -but I just have it set to "on" or "off" & it's so quiet it doesn't really matter, it just works.