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Quote from: Viva Cristo Rey
136. What is the Church?
The Church is the congregation of all baptized persons united in the same true faith, the same sacrifice, and the same sacraments, under the authority of the Sovereign Pontiff and the bishops in communion with him.
The kingdom of heaven is like a man who sowed good seed in his field. (Matthew 13:24)
137. Who founded the Church?
Jesus Christ founded the Church.
And I say to thee, thou art Peter, and upon this rock I will build my Church, and the gates of hell shall not prevail against it. (Matthew 16:18)
138. Why did Jesus Christ found the Church?
Jesus Christ founded the Church to bring all men to eternal salvation.
My sheep hear my voice, and I know them and they follow me. And I give them everlasting life; and they shall never perish, neither shall anyone snatch them out of my hand. (John 10:27-28)

In my opinion, the idea of "universal acceptance" has to do with the conclave and the election of a pope.  If the pope is accepted then he's the pope.  But...this idea of acceptance doesn't give the pope a lifetime of immunity or a papacy free of critique.  If a pope was elected and accepted (let's say, +Benedict) but then a few years later, he starts visiting Jєωιѕн ѕуηαgσgυєs and praying with infidels, or speaking heresy, then the doubts which arise have nothing to do with acceptance of his election but with acceptance of his orthodoxy.  Universal acceptance as a pope elect is different from universal acceptance as being a member of the church, if there is doubt as to heresy or whatever.  Most theologians couldn't wrap their heads around the circuмstances we live through, (how many of them thought a pope could never be a heretic...obviously there were very wrong) so for them, if a pope were elected, then that's that.  They never considered the possibility of a lack of universal acceptance due to a lack of orthodoxy.  But I think such a distinction is necessary.