He also strongly implied that a not-insignificant part of the old “law” were cultural traditions and compromises, rather than directives from god.

First see, this.
Matthew 22:34-40

Then this.
Mark 7:1-13

And then this.
Matthew 19:3-8

He was very clear that what he laid out in Matthew 22:34-40 was the summary of all law, and that any and all others must be derived from that. If they do not, it is just a tradition.