Given that the OP published the exact amount of BTC in question (0.6104), is it possible to search all addresses that contain exactly that amount in case it is only one address?
OP didn't mention the exact amount. He was only talking about an approximation:
a btc balance of 0.6104...ish.
This discussion reminds me of all those topics where people discuss what should be done with lost coins or the coins that people consider to be lost and irretrievable. The answer is nothing. The network cannot and must not be manipulated to put any coins back into circulation just because some of them are lost or people consider them to be lost due to the fact they haven't moved in a decade. If immutability is a pillar of Bitcoin, it has to remain that no matter what. You don't know if I have coins that haven't moved in since 2012. Maybe I am planning to keep them there for 30 years and it's my retirement. I don't have to prove to anyone that I am still here and those are still my coins. Bitcoin stops being Bitcoin with such changes.
If OP can't find the right owner of the paper wallet, nothing should happen with those coins, ever. Maybe the wallet belongs to someone who invested in Bitcoin and has no interest in checking his wallet or the blockchain for the next decade or two, and once he is ready to use them in 20-30 years, they should still be there.
@OP do your best to try to find the legitimate owner but don't move their coins.