I concur. When Jesus said “It is written” we can know for sure that this is God’s law. But when we read “You have heard it said” or “The law says” we can identify a distinct difference in that it is tradition that is being referred to. Of course not all tradition is bad, but any tradition that contradicts scripture is not to be held on to or trusted.

“the law says” is found 3 places in the NT, the word “says” is a clue that what MAY be being discussed is the so-called Oral Law of the Pharisees (later written down in the Mishnah), which includes the Written Law (what we call the OT), which in turn includes the Law of Moses (what we call the Pentateuch).
The point is that the “the law says” in ALL 3 cases in the NT can mean Paul is referring to the Oral Law, that is, to followers of rabbinic tradtion. Not all of their traditions negated the Written Law, just some of them. We do know that the Mishnah says women should be silent.

We know from the Dead Sea Scrolls that “works of the law” refers to the boundary markers that identify Jews from gentiles.