Cabin door hinge?.. Doesn''t your boat have hatch boards, or is this an internal door? I am not really sure I can picture where the problems exists. What kind of boat is this? (a daysailer?) what year? (is it quite a few years old?) Is there alot of torsional loading on these spots. I agree that the best thing if possible is drilling all the way through and bolting. But I bet that is not possible. You could try enlarging the hole a bit and filling with an epoxy filler that does not get too brittle, but has a little "rubberiness" to it (or it will just crack in the new hole.
I don''t know exactly where these screws are but on a lot of the newer "Mid-Level" production boats there are backing plates "embedded" in the fiberglass in areas of high stress (like where clutches are placed, etc). But I bet that in areas of low-stress/importance (door hing?) there isn''t any. It sounds like a poor mechanical design for the load that is induced on these areas. Alot of the newer bigger boats that have been in production for a while usually work these annoying design flaws out after a few years.
BTW, I agree with Denr in this is the area, that this is where you would notice the difference between a Hinckley and a "Mid-Level" production boat. Not necesarily in sailing charcteristics, but the fit and finish and small details that might make the boat just a little more of a joy to use. But then again thats why you pay Hundreds of Thousads of dollars more for the boat. But I think proper design would eliminate most issues. Though using self tapping screws on piano door hinges is not a "cheap and dirty way to put things together" Typically its the ONLY way to attach something. Its all in the design. I am sure I can go on a Hinckley today and find LOTS of self tapping screws, especially down below. Its just where they use them. I know in the 80''s some manufacturers where saving time/money by using self tapping screws on things like a Hull to Deck joint. Now that is where if I was beating the wrong way around cape Horn that I would rather have the beefier through bolting. But in some areas its the only way to go.