If you use an average of one 7 or 8 oz trichlor tablet for every 10,000 gallons of water, and replace them regularly the FC and CYA levels should track each other appropriately. When CYA levels are low, the FC level you use can also be low. Keeping FC at 3 when CYA measures as zero is not required.
The CYA test can't measure any CYA level below about 20, and will simply report lower levels as zero. If you are using tablets, your CYA level is going up and will eventually get into range.
CYA and chlorine levels can be a little tricky. The key to understanding is that FC levels and the amount of chlorine needed to maintain that level are two different things. At lower CYA levels you maintain a lower FC level, but need to use more added chlorine to maintain that level.Thus, at higher CYA levels you will be spending less on chlorine, even though there will be more chlorine in the pool.