Would that mean... if Congress desires to "get" he President, Congress should "get" him through their own activities?
Yes, if I understand what you mean by "get". The justice department can investigate executive branch employees, of course, but the president has the right to tell them to stop. Then they can resign, like Archibald Cox did.
Nixon was brought down by the executive justice department, because he couldn't keep firing the people in charge. It proved that good people can give us good government, but the dems used it anyway to push the independent council law.
The president can't control the executive power of the country if the congress can legislate a separate executive with executive powers that reports to them.
Note that the supreme court ruled the line-item veto unconstitutional because it gave congressional law-making power to the executive branch, contrary to the constitutional provisions. The same should have been true of the independent council law, but the judges just thought they whole idea of an independent council was so "right" that it didn't matter that it was unconstitutional.
It's easy to see the problem if you take it to extremes. If the congress can pass a law making the FEC, with law enforcement powers, which the president can't control, why can't they pass a law putting the military under control of an independent executive appointed by congress? Why not make all cabinet members appointed by congress rather than the president?
The court has, in typically activist style, decided to apply the constitution selectively based on how "fair" the outcome is -- so since it makes sense that the FEC should be outside of politics, they let congress set it up that way.
Well, I have a new theory. NOTHING should be outside politics. POLITICS is how we the people control our own fate. If we don't like what government does, we vote it out and vote in people we like. And the constitution protects the minority from government interfering with them too much because of us in the majority.
But the constitition makes no guarantees that parts of our government will be run without regard to who wins elections.