If the blockchain records all transactions, how does losing your wallet imply that you lose your Bitcoin?
You've answered your own question - "all you would need to be able to do is prove that that address belonged to you." To prove the address belongs to you you need the associated private key, which is secret and typically kept on your computer. If you lose your key you are no different than any other pretender to the address.

This is why you must back up your wallet file containing the keys, so should something happen to your computer you can still access the coins using your backup.

Remember, this is a decentralized digital currency. There is no clerk you can go to with some documents who can authorize that this is your address. You must prove ownership in a universally verifiable digital way - by using your private key to provide a cryptographic signature.