The head reverse works typically by having the gear under the head being driven by a toothed lever. That lever (whose teeth are meshed with the head gear) is typically then driven by some form of linkage. If the reversing linkage is not functioning, it could have been distorted from the earlier attempt to reverse while the head was locked (from the screw driver), and this distortion, or bend or whatever might be preventing that linkage from actuating the lever. If that isn't the case, then perhaps the reversing issue is a red herring.
Of greater concern is the issue where the main drive belt was routed wrong. This could have caused issues with the drive mechanism since much of the deck assembly is intended to only run in one direction. On most boomboxes, spinning the flywheel by hand in one direction runs smoothly, but if you try to spin it in the wrong direction, there is usually significant resistance and if you force it, an awful noise usually results. So while adjusting the tape with the incorrect belt routing, the heavy (reverse) flywheel was spinning in the opposite direction. Since it wasn't really being engaged at the time (standard direction), it may not have been an issue. However, the moment you hit reverse, that flywheel likely engaged the reels and other gear assemblies (with all the inertia of the flywheel) going in the wrong direction. Personally, I suspect a deeper disassembly is needed to expose the gears behind the flywheels to see what has occurred. Hitting reverse was a big mistake since you would only adjust azimuth with the head from the normal side. Once you get the azimuth angle correct, the reverse should be exactly 180* and still correct. Furthermore, although the diy folks here frequently recommend tinkering with the azimuth setting by "ear", the correct way to adjust azimuth is with an oscilloscope.
Of greater concern is the issue where the main drive belt was routed wrong. This could have caused issues with the drive mechanism since much of the deck assembly is intended to only run in one direction. On most boomboxes, spinning the flywheel by hand in one direction runs smoothly, but if you try to spin it in the wrong direction, there is usually significant resistance and if you force it, an awful noise usually results. So while adjusting the tape with the incorrect belt routing, the heavy (reverse) flywheel was spinning in the opposite direction. Since it wasn't really being engaged at the time (standard direction), it may not have been an issue. However, the moment you hit reverse, that flywheel likely engaged the reels and other gear assemblies (with all the inertia of the flywheel) going in the wrong direction. Personally, I suspect a deeper disassembly is needed to expose the gears behind the flywheels to see what has occurred. Hitting reverse was a big mistake since you would only adjust azimuth with the head from the normal side. Once you get the azimuth angle correct, the reverse should be exactly 180* and still correct. Furthermore, although the diy folks here frequently recommend tinkering with the azimuth setting by "ear", the correct way to adjust azimuth is with an oscilloscope.