Kissinger undermined Rogers at every opportunity because he was profoundly insecure. Kissinger would yell at anyone available (quite often Haldeman) for hours about slights, real and imaginary. The story of Kissinger’s time as National Security Adviser was a story of constant tussling with Rogers, and making sure Rogers was kept in the dark.
Nixon initially undermined Rogers so that the White House, not the Secretary of State, would have control of foreign policy, which was all that he really cared about. It’s why he deliberate chose a former business partner who knew nothing of foreign policy. Of course, once Kissinger became too big for his boots, Nixon found he needed to undermine Kissinger too, and very rarely he would side with Rogers against Kissinger.