PHILADELPHIA — The trust of the American electorate remains out of her reach. Its affections still elude her.

So after 40 years in public life, Hillary Clinton grasped for something less lofty and far more attainable here on Thursday night: an unquestioned acknowledgment of her readiness for the job.

The most consequential speech of her campaign, and of her career, seemed to contain an unspoken concession: By now it is all but impossible for Mrs. Clinton to shake herself loose from the elaborate, decades-old scaffold of assumptions — built up by scandal and bolted in place by her enemies — that her motives are impure.