The main thing I don't love about that inheritance rule, from a meta-gamey perspective, is that it's an extra layer of "planning ahead," which slightly contradicts the theme I associate with old-school play, "Your character will probably die, but that's fine: You can roll up a new PC in 5 minutes and jump back in the game." I'd just assume say, that any future-replacement-PC-to-be-named-later can crawl out of the woodwork, to claim the inheritance. Keep it loose and spontaneous.
One of the ideas I flirted with, when I was first planning this campaign, was an answer to the question, "What happens when your Hobbet PC reaches maximum 4th level? What then?" I toyed with the idea, that at that point, you would add a second PC under your control. So for example the same player would control both Frodo and Sam, as their primary and secondary PCs. (And if you got Sam up to 4th level, then you could have a 3rd, and eventually 4th, 5th, 6th Hobbet PC.) This would create a built-in "succession/inheritance" system, whereby for example, if Frodo dies, then Sam becomes the primary PC and takes up the Ring, and it stays under control of the same player.
The more I think about this, the more I'm leaning towards, I just want to handle it socially within the game. When your PC dies, their treasure goes in the party's "kitty" and the surviving PCs can scavenge through their equipment for useful items. When you roll up a replacement PC, you start with the standard 30-180gp, no more, no less. The other Hobbets, in the interest of building the most formidable team, might choose to help out equipping the newcomer (or not). Maybe the other PCs would give your replacement PC extra gold, to buy better equipment, or maybe they'd even pass down a magic item from your fallen PC. This is perhaps an incentive toward good role-playing of character death: If your PC goes out heroically, sacrificing themselves for the good of the group, then the other players would be more inclined to pass some of your old PC's items along to your replacement PC.
Finally, a deep thought about the difference between "treasure" and "money" in my campaigns: "Treasure" has a near-mystical quality, that it's a lost resource, that's been taken out of the economy and lessens the community. Buried pirate treasure, dragon's lair, mummy's hoard. That's why you get XP for finding it and bringing it back: you are boosting the economy and benefiting Lawful civilization. But once the XP is awarded, the mystical quality is drained, and it just becomes "money." So two different PCs can't get XP from the same treasure, for example, I would be disinclined to ever award XP for inherited funds. It's not "treasure" any more at that point; just "money." (The exception is if the money gets taken out of circulation by bad guys: for example stolen by bandits. At that point it becomes "treasure" again.)