Re: Probably different model
Absolutely.
Bitcoins are basically generated by an anti-spam measure to stop one person hoarding all the coins without a proportional expense/effort on their part. You could have literally let people get a Bitcoin just by sending an email to an automated address, if you'd wanted, and used that as the ledger entry.
A ledger itself, though, just needs to know who made the transaction, who signed off on it, who can revoke it, and then put it into the ledger. Importantly, even complete compromise of their key cannot erase the history of what that key did, as each transaction is in effect "signed" by the next (basically the core idea of blockchain).
Think of it not like Bitcoin but like those play-by-mail games of old. You collect the transactions, perform the calculations, and modify the "database" in the middle. Then you send out the results of the next "round", everyone else can see the effects of those new results and "previous moves" but you can't change them.
The only difference is that instead of one person performing the calculations, you're publishing the data, letting everyone do the calculations at the same time, waiting for everyone to agree what the result is, and then having everyone sign off on that result.
Bitcoin does that, WHILE solving a pointless maths problems that's hard to solve (I believe it is literally "hashing the entire existing blockchain with random numbers until the hash results in 00000000000000000000000" - or it certainly was at one point). It's literally a time-effort-and-money-burning exercise so that there's something that has to happen to make a coin.