Im not a lawyer and I don’t play one on TV, but there’s a lot more than “follow the law” packed into that little paragraph

Not really. Boiled down, the oath involves an affirmation to follow and protect the Constitution and to do their duty under the law. Since the Constitution is also a law, there's really not anything more there than "follow the law."

The Constitution doesn't say anything about DOJ being independent. And the law doesn't say anything about the DOJ being independent. It's not a legally an independent agency. By tradition there has been an expectation that the DOJ would operate somewhat independent from the White House, but that's not a creation of this oath.

Every other cabinet official takes the exact same oath, and the expectation is that they will follow the policies and directives of the President. None of those other agency Secretaries are expected to exhibit "independence" from the President the way that DOJ has traditionally been expected to do so. But whatever the source of that expectation for the DOJ, it isn't from that oath.