Gotoh Mini 510 Tuning Machines

johnnyguitar, that is a great guitar.  Basically a Seagull S6 classic with the thinner nut width and the Q1T pick up.  One of Godin's best acoustics in my opinion.  Great bang for the buck.

terryd , all the broken screw extractors I've seen would result in a larger hole once the extraction is performed. The hole could then be filled in and re-drilled though.
I know what you mean about the Seagull tuners. The tuners on my Maritime are not consistent - one will be real stiff turning, and the next one easy.
The open gear tuners on my NE are better in my opinion.

My original post was quite a while ago and the glued screw has held to this point and these tuners will never be changed as long as I own the guitar so I'm thinking the problem is fixed.  It is disappointing that a company like Gotoh doesn't provide better quality fasteners though.  That had to be pretty soft metal to break off with no more torque than I was applying.

This is just a pondering, but it might not be that the screw is bad, but that the screw is so tiny.  I mean, you've almost got a piece of threaded wire there.  On top of that, you're screwing into hardwood most of the time.  lol.

Pre-drilling an even tiny-er pilot hole is a good idea imho.  May seem like a lot of trouble, but in the case of putting new tuners on a guitar, one would think you'd only be doing that once ever.  So the old, "twice measured, once done" adage would probably apply here.  Otherwise, one might be singing a different tune.    (see what I did there?!!) 

terryd , I've broken many screws over the years, but not on guitar tuners. I've set many tiny screws as well, on very small hinges when making tiny boxes, that sort of thing...a lot of them are brass, to prevent corrosion, and then so small - the brass isn't very strong when it's so small. Don't know if the Gotoh screws were brass or not....

StewMac has a couple of videos on repair of these broken small screws. It does involve drilling out the broken screw, creating a larger diameter hole in the process, then plugging this hole and finally re-drilling a new hole for the screw. Dan Erlewine makes it look easy, as always. They sell a tool kit (mainly specialized drill bits) to accomplish this task.