Synchronizers are friction devices. They are most commonly made out of brass although other materials may be used. Brass is a soft metal and as it wears down the power it produces would not damage other components.
The synchronizer resembles two cones that fit one inside the other. One cone is attached to the blocking ring and the other to the gear train. In neutral none of the synchronizers are touching. As you begin to shift and the blocking ring moves toward a gear the first thing to make contact is the synchronizer. The synchronizer inside the blocking ring is spring loaded and as more pressure is applied to the shift more pressure is applied to the synchronizer. When enough pressure is applied the synchronizers stop slipping and start rotating as a single unit matching the speed of the great train and blocking ring. Finally the blocking ring engages the dog teeth of the gear train and the transmission is in gear.
The metal chosen for the synchronizer, the surface area that touches inside the synchronizer, and the pressure of the springs are all factors that are chosen keeping in mind that everything will be swimming in gear lube. Not only does the gear lube provide lubrication it also provides cooling by transferring heat from inside the gear train to the transmission outer case.
If 5th gear was at the top of the transmission then it would go dry first if the transmission has a slow leak. At first the increase in friction from lack of gear oil inside the synchronizer would improve performance but at the cost of increased wear. Also with a lack of oil inside the synchronizer cooling would be diminished further speeding up the wear process. If 5th gear is at the top of the transmission it may have failed by now and refilling the transmission with oil may not bring it back.
To further illustrate the importance of the correct friction inside the synchronizers. I worked on the manual transmission where the customer added Slick 50 oil additive. The transmission began grinding in every gear. The synchronizers could not do their job because the friction inside them was reduced to such a low level. Several drain and fill of carb cleaner and a refill of clean gear lube fixed the problem.
Specific to the transmission of the original poster.
According to the parts explosion that was provided the transmission has two shafts, input shaft and main shaft. The main shaft has two blocking rings. One of the blocking rings serves 1-2 and reverse (how it also serves reverse i don't know but i'm willing to bet that it has something to do with the gear that's built into it). The other blocking ring serves 3-4. The final blocking ring is on the input shaft and is labeled OD (overdrive) this is 5th gear.
Looking at the picture provided in the post, the shaft in the foreground the main shaft as indicated by the two blocking rings. The shaft that's peeking out from behind the main shaft is the input shaft. The shift fork can be seen on the left. Extrapolating the orientation of the transmission the input shaft is the highest shaft in the transmission. This makes it plausible that 5th gear would go dry first damaging the synchronizer.