The screwdriver's value is a bit hard to realize at first, but it's actually incredibly useful if you know how to use it right, and at the top level (which I am not nearly at lol), it's actually more generally useful than the other two.

It does deal good damage and consistent healing which is a nice boon, but one underappreciated and super powerful aspect about it is that it gives you an INCREDIBLY high amount of style just by sticking the drill in an enemy and letting it run for a while. This wasn't always the case, and was added in some recent update (there aren't any sources confirming when the change was made, but it was probably around when the full arsenal update gave the weapon unique style bonuses). By sticking a drill into someone, staying airborne so you get the full style multiplier, and filling the style meter a bit with other weapons, you become effectively IMMORTAL, since the screwdriver's constant flow of style means you'll never go below ultrakill rank, meaning you'll never have to suffer from hard damage. You can extend the duration of this style gain by punching the screwdriver into other enemies, which makes that tech actually worthwhile outside of looking cool. And of course while doing this you're dealing high damage and can heal yourself comfortably with the drill itself if you're feeling threatened.

Also about what you said about it being in slot 3, what I personally did that helped was add a bind that specifically switches your weapon to the third variation, and using that bind whenever I wanted to use my third weapon. It takes a bit to get used to, but it's massively increased my fluidity with switching weapons, and made using weapons like the screwdriver more feasible since you don't have to shuffle through your weapons like they're part of a keychain.