Reduce, Reuse, Recycle

Question: When a Scion dies, what happens to his Birthrights? is it possible to make a Scion who has the concept of hunting lost Birthrights/Relics and reclaiming them and giving them to other Scions for re-use?

Technically, nothing happens to a Scion's Birthrights when he dies. He's dead, but the objects aren't; they're still there with his body, dropped and forlorn. They could be damaged or destroyed if whatever killed him was sufficiently dangerous enough to damage relics - for example, he fell into a volcano or something - but most of the time, they'll be just fine.

As a result, anybody can pick them up and make off with them after the Scion meets his unfortunate demise. Quite often, whomever killed the Scion will take their relics as trophies, part of the spoils of battle. If the enemy who took a Scion out can't identify relics, doesn't want them or is too pressed for time to bother with them - or if it wasn't a being that killed the Scion but bad luck or environmental hazards - they probably remain where they fell. Relics have their own weaker Fateful Auras, however, and are hot commodities among gods and Titans, so it's unlikely that too many relics remain unclaimed for very long; even if an enemy doesn't take them, whatever divine creatures are nearby, not to mention the Scion's parent, probably will.

However, sometimes relics do lie around for a while before someone suitable comes along to find them, and sometimes they're in such challenging locations that it's not worth it for those who aren't dedicated to go retrieve them. In those cases, a Scion who wanted to make Birthright retrieval one of his priorities could absolutely go forth and try to grab them up, braving danger, turmoil and others with the same goal.

We've actually had Scions pick up lost Birthrights in our games several times; the relics belonging to Wolf and his followers before they died were found by Aurora's band and later became major plot motivators, and both Sverrir and Folkwardr were gifted by gods with relics that once belonged to the Jolie-Pitt kids (Sverrir gained possession of Darcy's Dodge Ram, while Folkwardr was given a piece of Carlos' corn-husk shield) while Sowiljr once picked up a sword that originally belonged to Ian Jupiter. Some gods that famously died in the World also left behind major relics, and the PCs have often wished they could get to the bottom of the Wyrd-infested Pacific ocean to retrieve the magical skis and bow left behind after Uller died there at the onset of Ragnarok.

A Scion who wanted to become a sort of relic-recycling fairy could definitely make it her business to run around trying to get ahold of these relics to make sure they get rehomed with new, deserving owners. It would likely be harrowing, since many other people would want them, and there would also be a whole set of challenges associated with trying to give them to others; she'd need to manage relics that might bite her, have the higher enemy-count that carrying a whole bunch of relics always brings, and probably need some serious diplomatic chops to deal with a lot of gods being very, very pissed that she's decided she gets to redistribute the relics that by rights they created or believe belong to them. She'll also probably want to be a badass enough mage to use Birthright Bond to give those relics officially to whomever she's handing them out to, to ensure that they're able to actually use them (although if she's not high enough Legend to get it or just doesn't want to go the Magic route, it's also pretty hilarious for her to give out relics that have a chance to blow up on the lucky Scions who get them; nobody said fancy prizes were free).

But heck yeah, if you want to create a Scion who makes treasure-hunting and Robin-Hooding old relics one of his major foci, that's totally doable. Just be aware of the pitfalls, invest a lot in tracking, hunting and/or Magic, and be ready to dare the exact places that are most lethal to previous Scions.

All of the above regards relics, which it sounds like your question was mostly referring to; things are more nebulous for non-item Birthrights like Followers and Creatures. We usually assume that those either die with their Scion (but could be brought back with him if he were to return to life) or are able to keep running around and doing things only as long as they can survive on their own. Just like relics, they would be uncontrollable by any other Scions unless they were severed from their original owner and rebound through the use of Magic.