In that video, Penrose explicitly claims that quantum mechanics "is wrong". He means that the equations of quantum mechanics are not sufficient to describe the universe. He thinks you need some additional mechanics to cause wave function collapse.
Wheeler, on the other hand, does not claim that quantum mechanics is wrong or incomplete, but suggests an interpretation of the equations. So in my mind they are taking very different approaches.
I watched the later part of the video you linked where Penrose describes the thought experiment with the planet without any conscious agents. He describes wave function collapse as an objective process that is "caused" by conscious measurement. Whereas my understanding of Wheeler's ideas is that wave function collapse is a subjective process.
My recollection of Penrose's quoting in that video is that he said 'Physics collapses the wavefunction' implying he believes it is not a consciousness phenomenon.
Wheeler, on the other hand, does not claim that quantum mechanics is wrong or incomplete, but suggests an interpretation of the equations. So in my mind they are taking very different approaches.
I watched the later part of the video you linked where Penrose describes the thought experiment with the planet without any conscious agents. He describes wave function collapse as an objective process that is "caused" by conscious measurement. Whereas my understanding of Wheeler's ideas is that wave function collapse is a subjective process.