I used to base it around St. Athanasius' explanation that the Church (at the time I would have believed the RCC) is the 'throne' of Christ, in which He resides. Where the Church is, as is Christ. So when Pope Leo XIII said "But since We hold upon this earth the place of God Almighty, Who will have all men to be saved and to come to the knowledge of the Truth", I believe he was speaking of Christ's Church, His throne, where men are to be saved and Divine Truth found.
And before anyone calls me a crypto-RC, LoL, that is how I would have read and understood the Letter when I was a RC.
This reminds me of a line from the Patriarchs response to Leo's predecessor:
Usurping as his own possession the Catholic Church of Christ, by occupancy, as he boasts, of the Episcopal Throne of St. Peter, he desires to deceive the more simple into apostasy from Orthodoxy, choosing for the basis of all theological instruction these paradoxical words (p. 10, 1.29): "nor is there any reason why ye refuse a return to the true Church and Communion with this my holy Throne."
10. Each one of our brethren and sons in Christ who have been piously brought up and instructed, wisely regarding the wisdom given him from God, will decide that the words of the present Bishop of Rome, like those of his schismatical predecessors, are not words of peace, as he affirms (p. 7,1.
, and of benevolence, but words of deceit and guile, tending to self-aggrandizement, agreeably to the practice of his antisynodical predecessors. We are therefore sure, that even as heretofore, so hereafter the Orthodox will not be beguiled. For the word of our LORD is sure (John x. 5), A stranger will they not follow, but flee from him, for they know not the voice of strangers.
11. For all this we have esteemed it our paternal and brotherly need, and a sacred duty, by our present admonition to confirm you in the Orthodoxy you hold from your forefathers, and at the same time point out the emptiness of the syllogisms of the Bishop of Rome, of which he is manifestly himself aware. For not from his Apostolic Confession does he glorify his Throne, but from his Apostolic Throne seeks to establish his dignity, and from his dignity, his Confession. The truth is the other way.[/
And lest this be seen as divisive, they also state:
But if his Holiness had sent us statements concordant and in unison with the seven holy Ecumenical Councils, instead of boasting of the piety of his predecessors lauded by our predecessors and fathers in an Ecumenical Council, he might justly have gloried in his own orthodoxy, declaring his own goodness instead of that of his fathers. Therefore let his Holiness be assured, that if, even now, he will write us such things as two hundred fathers on investigation and inquiry shall find consonant and agreeing with the said former Councils, then, we say, he shall hear from us sinners today, not only, "Peter has so spoken," or anything of like honor, but this also, "Let the holy hand be kissed which has wiped away the tears of the Catholic Church."http://www.orthodoxinfo.com/ecumenism/encyc_1848.aspx