Those tiny screws on Vintage Tuners are an issue for lots of people...and have been for many years.  The screws that come with the Gotoh tuners have the same issues.  This seems to be a common issue with all manufacturers.  But, the screws are so small that no tuner manufacturer seems to have found a better solution yet.
 
In most cases, the tiny screws will install easily if a couple of precautions are taken.

1.) Drill a pilot hole for each screw.  Without a pilot hole, the wood may crack or the screw head will break off, or both!

2.) Apply bar soap or candle wax to the threads of the screws.  This reduces friction.

3.) Use the correct size Philips screw driver, so that the screwdriver doesn't slip or damage the screw head.

4.) The screws should mount smoothly.  If you feel resistance or binding or a lot of friction...STOP!  Something is wrong.

These same four precautions should be used for every screw that gets put into guitar necks and bodies.