Ledman - Gillettes are fun. That series was called the "Gillette Reflex". The movement was made by La Generale Corp. the tip feature is nice. Your perception is correct - tip watch to about 45 degrees, after 1 second of non-movement the display comes on for 2-3 seconds. Pressing the time button can only give time, seconds and date(numeric). It sounds like the same, single wire bond is broken or disconnected.* There are a few of us who can attempt the repair...there is the possibility it is in the actual circuit, not just a broken connection. It is a very fragile display, so I personally seal those displays permanently with clear slow-set epoxy when done working on them. Never to be fixed again, but never to be broken again either(probably).
*The same connection powers the same segment of ALL the display charecters. The charecters are actually taking turns being on(and off), very fast, sort of like motion picture frames giving the appearance of motion. The process is called "multiplexing". http://en.wikipedia.org/wiki/Multiplexed_display