In 1988, Kinnock was challenged by Tony Benn for the party leadership and he appeared mired in internal battles after five years of leadership and the Conservatives still dominating the scene and ahead in the opinion polls.
Which is why he won't leave to fuck off to America to be on the Cabinet of some guy he barely knows.
Kinnock was pulling his party back together. He felt it his duty to expel Tendencies, beat off any attempt by the Bennites to bring the Party back to 1983, and make it a fighting force that could oppose Thatcherism and win an election. The poll numbers don't matter in 1988, because for Kinnock the struggle was one of an existential threat facing the movement he was leader of, and one he would see until he himself could no longer fight, or if someone was able to continue his work. He is not, would not, and could not fuck off to America, as this would compromise his principles and morals, as well as his duty to the Party and the movement.
EDIT: To also address another thing, even if we accept the premise 'oh Kinnock was just waiting to drop out', which is bullshit in itself, the moment you bring Benn into the picture, Kinnock will
not leave. Kinnock, for everything, will not let the last five years of struggle be reversed. He will not allow the man who wrote the longest suicide note in history, and thus bring the worst results the Party had seen since the 1930's, be his successor- he will either face him in a Leadership election and win, or face him and loose.