What is the purpose of the ledger? Given that most wallets are anonymous, what is the benefit of maintaining the history of all Bitcoin transactions?

Bitcoin is a payment method designed to circumvent regulation. When internet commerce began, it was clear that existing payment methods were not working. A new system was invented. E-Gold was the most popular internet payment system, but it was the target of allegations of money laundering and operating unlicensed financial services. The founders had to accept a settlement and shut down.

Bitcoin was not designed to be a business or a company with a boss. The Bitcoin blockchain is not a copy on a server. It is made up of thousands of copies run by volunteers in over 100 countries. This makes it impossible to turn them off. It continues because there is no boss to attack them with criminal charges, no boss to force them to shut down.

Bitcoin wallets are not accounts with balances. Bitcoin wallet balances are not stored on Bitcoin nodes. Bitcoin's blockchain only stores transactions.

If I want to buy a pair of shoes on the internet, how does the shoe seller know that the coin I'm spending is valid, i.e. that the coin exists, that I haven't spent it yet, and that I am spending it? Am I the owner of the coin?

He knows that I'm the owner because the signature of my transaction matches the address stored on the coin in an old transaction. He can't know the address unless his own Bitcoin node can see the old transaction, i.e. the transaction that created the coin when the other person sent me Bitcoin. He can't know that the coin exists unless he can find it in his copy of the blockchain. He can't know that the coin hasn't been issued unless he can find it in his node's UTXO database.

How does the shoe seller check that my coin in his copy of the blockchain is the same as the one in his copy of the blockchain? He can't do that. He can only check all transactions in each block to see if each block is valid and if the hash in the block header correctly links each block to the previous one. If his node has attested to the blockchain from creation to present and my node has attested to my blockchain from creation to present, then I can send him my coins to buy shoes.