An 8130-3 is simply a different form of a log entry. New parts frequently come with them too, and it will say "NEW" or "MANUFACTURED" in block 11. For units that were serviced in some way, "MODIFIED", "TESTED", "ALTERED", "OVERHAULED" or "REPAIRED" are common. If you sent it out for a battery replacement it would likely be called "REPAIRED" with the battery replacement listed in block 12.
If a unit is sent out for a failure, and nothing is found it usually comes with the 8130-3 saying "TESTED" and in block 12 it will list the maintenance data and task they performed during troubleshooting and "unable to duplicate discrepancy".
If you are selling overseas they may require tractability in the buyer's country. Depending on their regulatory agency it may even dictate that it must be traceability issued by an OEM approved facility, meaning an 8130-3 filled out by limited FAA repair station # XXXXX, (Joe Blow's Resale avionics) isn't acceptable.
The 8130-3 only documents that whatever actions listed in block 12 have been returned to service. The part can still show up DOA or damaged in shipping, so the 8130-3 itself can be just a useless piece of paper.
If we are talking prop overhaul, and you are researching AD's, that 8130-3 or yellow tag can be a big money saver.
Some shops require all parts to have traceability like that, it may be even written into their Repair Station General Operating Manual (an FAA approved manual describing how the facility must operate to maintain the repair station certificate.) So they would require it while Joe Blow's Fixing wouldn't.