Bose car amp burned part
My question is: with the further information about the part (diode?),
will the amp work if I put it back in my car, albeit without
protection, with the component removed, as assumed before?

If it is (was) a reverse protection diode then there is no reason why the amp shouldn't work without it, provided it did its job and the rest of the amp wasn't damaged. However, without that protection the amp is vulnerable to future 'events'. You should replace the diode with one that has a suitable peak current rating, preferably a Schottky type for lowest voltage drop.
In case there was other damage, it would be prudent to test the amp with a current limited power supply, a lower rated fuse, or an incandescent lamp in series, to limit current if some other damaged part causes a short circuit.