try to connect the + 5V terminal of the L298 module to the + 5V pin of the Arduino. and then test that the engine is now running.
Have you ever tested whether the engine runs when you connect it directly to the battery? The battery you are using is not actually intended to power motors, it has enough voltage but not enough power. At least it won't last very long, but it should be enough for short tests if the engine is running at all.
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