Okay, stop. The 10-32 screw requirement is for the lower screws which attach the fan to the box, not the upper screws which attach the box to the wood. You can see the holes that the 10-32 screws go into, in those tiny squares to left and right of your photo.
It's entirely possible those squares do something clever, like capture a nut. But they are designed to do a good job taking a 10-32.
I don't know if the kit comes with special, hardened #10-32s. Suffice it to say I bet you haven't lived for years with this mess on your ceiling, so the magic screws probably secured the old lamp. Go get them back. If that means digging in the trash, let that be a lesson not to throw away things before the job is done.
Other than that, what makes them "magical" is probably their carefully chosen length.
DO NOT REPLACE THE DRYWALL SCREWS WITH 10-32
The 10-32 requirement doesn't refer to those!
These screws need to be strong enough to carry 16 pounds and the box. Drywall screws will do the job, but I would convert to deck screws (a wee bit thicker) if they are the same thread pitch.
Lay the screw threads so they touch, and see if the threads have the same spacing. Oh yeah, feel free to unscrew one of the drywall screws at a time to evaluate it - not both at once. The stiffness of unscrewing it will also tell you something about how thick the wood is. If it gets easier as you back it out, that tells you the wood is pretty thick.
Alternately, I would pre-drill small holes in the two remaining locations, and fit the deck screws there. You must pre-drill! The deck screws will certainly go in if you don't pre-drill, but they will likely also split the wood right down the line of the screws - and then none of the screws will have any gripping power at all.