Yhwach could stay Soul King if some sh*t happened to him that made him significantly weaker. You know the story "Sealed King of the quincies blah blah blah in 999 year's hes gonna f*ck us up" that was talking about Yhwach after Yamamoto beat him. If whatever happened to seal him were to happen again, only permanently this time, The Soul King has been effectively replaced.
Aizen's exact motives for replacing the Soul King were unknown, but if Yhwach couldn't be sealed, then Aizen would be the next best thing. Although he would have to have some kind of seal put on him as well, since Chair-Senpai isn't going to last to the end of the series.
Ichigo isn't going to become Soul King, because he is a generic teenage protagonist who is suddenly thrown into a world full of supernatural sh*t, which he overcomes, before eventually offered a position of great power. And then, as the generic teenage protagonist he is, he will decline the offer, just like literally every other generic teenage protagonist in fiction. In other words, Plot.