Imagine a peer goes offline and stuff happens on the network while its offline, when the peer goes back online what state does it follow, remember that it is a decentralized network what peer does it trust? This is where the chain of verifiable transactions come in. Also with this setup it is kind of easy to prevent double spending.

> BTW cash does not have a recorded Tx history

Sorta... The central banks know every coin that has been created and notes that have been printed. When you fill in your Tax forms you are creating this Tx history manually.