That makes perfect sense. And I see that the prosecution of bribery is seen as not necessitating inquiry into legislative acts or motivation.

2. The prosecution of appellee is not prohibited by the Speech or Debate Clause. Although that provision protects Members of Congress from inquiry into legislative acts or the motivation for performance of such acts, United States v. Johnson, 383 U. S. 169, 383 U. S. 185, it does not protect all conduct relating to the legislative process. Since, in this case, prosecution of the bribery charges does not necessitate inquiry into legislative acts or motivation, the District Court erred in holding that the Speech or Debate Clause required dismissal of the indictment. Pp. 408 U. S. 507-529.

The bribery itself is the motivation. But would an aid's or advisor's notes and testimony be able to be summoned by the DOJ in the case where a killing did take place, and there is testimony that the order came from the President or one of his aides? I would think at that point the immunity would not function. I realize it's a hypo, but what if that's the link you need to establish?