This was continued (and probably expanded) in the Bush Administration, where Cheney's chiefs of staff also held the rank of advisor to the President.
But this integration has reached a new level in the Obama-Biden administration. First are the number of people with strong links to Biden in the West Wing. But also, according to the invaluable Plum Book, besides Biden's chief of staff, there are five Biden staffers who also hold the rank of Deputy Assistant to the Vice President (including the chief of staff to Jill Biden) and two more who are Special Assistants to the President.
But a picture tells a thousand words. In the now iconic picture of Obama and his team in the situation room monitoring the raid that killed Osama bin Laden, indicated in the picture with a big yellow circle and arrow (helpfully added with the app Penultimate) is VPNSA and Deputy Assistant to the President Antony Blinken. He may not be in the front row - but he is in the room. And in government, once a position/procedure/protocol is established it goes on forever.