The two small terminals are energized by a relay when you turn the ignition key. That makes the solenoid close the contacts between the two big terminals and supply power to the starter motor. So, you have already effectively jumped the solenoid by shorting the large terminals with the screwdriver.
If the starter motor just spins, you either have a) a trashed flex plate/flywheel with several missing teeth b) a very low battery or poor connections causing low power to the solenoid/starter motor c) a worn out starter motor that’s not spinning fast enough to extend the bendix gear or d) bendix is sticking and not extending properly.
A load test can tell you if the battery is low.
Next would be to check connections at the battery and starter motor.
Next I’d pull the starter and inspect the bendix gear and assembly. I’d probably bench test it too. If you’ve never done this, it has to be seriously restrained, there is tremendous torque when it is powered up. If the gear looks chewed up, that could mean the flexplate/flywheel teeth have been taking a beating too and need inspection. An auto parts store may be able to test the starter motor for you and make sure the gear is extending under power. Call around.