And yet, in many ways, if not for all of these factors — the ones that compelled Democratic voters to put aside the notion of nominating a newcomer in favor of a familiar, even comforting presence — Biden may well not have gotten this close to achieving his goal at all. To those who know him best, including some who have been by his side for decades, Biden’s improbable ascent so late in his life and during this unsteady period for the country can at times feel less like a man seizing the moment than the moment finally coming around to him.