Another word for Testament is Covenant. We read about the covenant in the Old Testament and it is for this reason it is sometimes referred to as the Old Covenant. A covenant is a contract. This can be easily verified. In every contract there exists terms and conditions of that contract. In the Old Testament, an example of this are the blessings and curses for Israel not upholding the laws and ordinances therein.

So, we are dealing with a great contract here. When you say you don't want to "get lost in legalism", the statement is ambiguous to me. A contract is ab initio a legal agreement between parties. A more accurate term for the New Testament is Renewed Covenant. The reason is that not one iota of the original contract (Covenant) was changed. Jesus' entire mission only changed the terms and conditions of this great contract (Covenant). Also, when you said, "the whole point of the new testament was that Jesus raised the bar,.....", I find this to be somewhat cryptic in meaning and opaque to me. Jesus came to fulfill prophesies of the Old Testament. And what is meant by "fulfilling the law" is fulfilling for all of us who believe in Him, the requirements of the Old Covenant to the exact letter and thus meeting the terms and conditions of the contract. The Old Covenant presented the Israelites with an impossible task of keeping the Old Covenant without any violations. Only Jesus succeeded in doing so. From his sacrifice, he paid for all our violations and as a blessing for His accomplishment, the Covenant was renewed with new terms and conditions. We paid with His blood so that we could live and have a chance of being saved.