the lesson leaves out both chambers ability to enact regulations
What’s true as a matter of Constitutional Law and what’s true as a matter of House Or Senate rules may be different.
It’s undeniable that the Senate rules require the Senate to conduct a trial, and to do so using using members of the House as the prosecutors. And that they also require the impeachment be transmitted to them by the Senate.
But it’s also undeniable that the Constitution itself imposes no such requirements. And that it makes no reference to any required “ministerial actions.”
If the matter ever comes before SCOTUS, that entity is duty bound to hold the Constitution’s mandates and constraints above the rules of either chamber of Congress.