I have zero trust in my bank but every transaction will not need to be verified throughout history.
No, that's not true. You have 100% trust in your bank to hold your money for you. The bank says "you have this amount of money", and you believe that number. You don't control the money, the bank does.
On the other hand, Bitcoin *is* zero-trust. If you run Bitcoin Core, you're validating every piece of transaction information that you receive, and not trusting or believing anything just because someone tells you it's correct.