Couldn't the blockchain prove to be basically proof of shipping?
No. Bitcoin transactions are anonymous and by a wallet ID/address one cannot prove a transaction as the wallet does not hold any proof like ID card or address proof. Only the IP address is recorded which is not sufficient that you have sent the bitcoins to the person. Xapo uses mobile verification but still it is not a solid proof.
That's why people prefer using Virwox or localbitcoins for exchange. Even Ebay transactions can be chargedback.
You can prove you own an address by signing a message using that address. The proof is the fact that you own the private key to that address which, like the name says, should be private and one that only you have. Also, your IP Address is never recorded.