Batteries can be consumable after a fashion, as they lose capacity with each charge cycle. The effect can be reduced via certain techniques (depending on the battery chemistry involved), but so far never completely eliminated. Just consider it entropy in action, really.

Is there any way to reclaim the resources in a "used up" battery to reintroduce them to the production chain?

You absolutely can, but it's currently not cost-effective. On the other hand it looks like it's cheap enough that making it mandatory or putting a recycling deposit on batteries would be viable.

Are battery recycling programs near you not feasible? Our local Home Depot and Best Buy stores will take back rechargeable batteries for recycling for free.

Do they recycle all of a lithium ion battery?

Very little of the physical (the lithium). However all of the precious materials (https://en.wikipedia.org/wiki/Battery_recycling#Lithium_ion_...).

It's not worth it to extract the lithium, monetarily speaking.