Anyone know how much of a headache it would be to just lease an aircraft short term for these kinds of flights to avoid having a specific jet tied to his identi

Anyone know how much of a headache it would be to just lease an aircraft short term for these kinds of flights to avoid having a specific jet tied to his identity? I'm assuming it's a logistical issue more than a financial one, or else more people would do it. (or private jetters can't abide to not have their own aircraft, always available and set up the way they like it)