My guess is that when the President is looking for plausible deniability for very sensitive matter, the cutout doesn't' use a public line with other people listening in.
Your definition of a public line may not be accurate.
Do you think Crown Prince used your version of a public line when he ordered the hit that Gina Haspel may or may not have said was recorded.
I don’t know anything about the Clarion Project. Some or none of the following story may be true
https://clarionproject.org/revealed-khashoggi-was-a-qatari-asset/
But is shows exactly why a Head of State would inform another Head of State.
Btw, Imam of Peace is hot and heavy on Qatar undermining the US including through Ilhan Omar. There is a current law suit over the ordered hit of someone that may bring out more details.