I believe President Packer means it when he says his testimony was "the same as I might have in a fast and testimony meeting in my ward." When someone in a position of Church leadership has an audience with Christ, we hear about it. Joseph Smith told us. Oliver Cowdrey told us. Sidney Rigdon told us. So did President John Taylor, President Joseph F. Smith and David B. Haight. Their calling is to bear a witness of Him. When they have an actual audience, I believe they tell us.
The calling of the Twelve is to "bear witness" of Christ. (D&C 107: 23.) Because of that calling, they must proclaim they have a "witness" even if it could be more correctly described as a testimony born of the Spirit. I accept their "witness" of Christ and believe it is authoritative. However, I do not read into their testimony what they do not put there themselves.
I accept the "witness" of the living Apostles, although it is a rare exception when one has an audience with Christ. In recent talks Elder Scott has gone to some length to testify and describe his own spiritual experiences. I trust in them. I trust him. I believe him to be an Apostle. It is not necessary for an Apostle in The Church of Jesus Christ of Latter-day Saints to have a personal audience with Christ.
Years ago Elder Mark Peterson said he did not think it possible for a gentile to receive an audience with Christ. He thought that was confined to pure-blooded Israelites. Since he was a gentile apostle to a gentile church, he did not believe it possible for him to receive such an audience. As I understand it, that is the general view among the brethren. The charge given by Elder Oliver Cowdrey to the Twelve (telling them they must receive an audience with Christ for their ordination to be complete) was discontinued in 1911 by President Smith. It was discontinued because so few had received that audience. But that does not make these men any less apostles.
I trust President Packer. I accept his testimony. I believe it is enough to qualify him for the work, just as President McKay told him. I am impressed with his humility in explaining his testimony in General Conference. It increases my trust in him as a servant of the Lord.