They controlled for exactly what you are saying. They added a key which has value only to the wallet owner and not the finder. There was not as big of an increase in returns. This led them to conclude the more likely explanation was the “I’m not a thief” reasoning.

I don't think the key is a strong control without more information about the key.

Say I've lost both a house key and also enough information to, in the current era of scummy personal information aggregation websites, find out my home address, which you can definitely commonly do based on just name and email address and an assumption of local residency. Then regardless of whether I get the key back I should in paranoia change the locks on my house, because now an unscrupulous person who found it easily has a copy of the key and knows exactly where to use it.

So if it's a house key, then in defense it should have no more value to me but has significant (hopefully temporary) value to them.

I'm not sure if the key is like that. In today's age of easy key duplication, you probably have key backups via your relationships and you could go to a key duplication vending machine and get it done for $5 when you go to a grocery store.

I'm quite impressed how in most countries, it's a function of empathy vs. how much hassle it is to return things.

The key value is in avoidance of loss (burglary), not in needing to get a new key cut.

Losing a key means changing the locks, the more duplicate keys the higher the cost.

Except that getting the key back still means changing the locks because the finder could have duplicated it, so it seems like there's not a lot of value in giving it back except in the case that it's their only copy that they will use until they change the lock.