Why is pulse important in collaborating with another musicians? Because it SECURES alignment. Without a common pulse between solo performer and collaborative pianist, you can be sure that alignment is going to be a nightmare. And that’s an understatement.
It is one thing to FEEL a pulse within oneself: it is another to depend on the mechanical, stalwart beat of the metronome.
I agree completely, although I work with my own students to help them physicalize the beat at the same time that they check it with a metronome. After all, we're only a few millenia removed from our cave-dwelling ancestors who probably started the musical impulse in the first place by discovering the primal sensation of beating on a drum.
I also keep a pair of bongos in my home studio (brought back from Jibacoa, Cuba) for pianists and singers to use in order to learn rhythms independently of their instruments. Another useful exercise is for the singer or instrumentalist to drum their solo line while I play the piano part. Once they separate the rhythm from the rest of the playing process it's much easier to keep the pulse when returning to their instrument.
How do you engage with ensemble partners in order to develop a common sense of pulse?