Here you present an article of a Catholic Encyclopedia of 1913. That's not the Magisterium of the Church, it has no authority. I won't be convinced by such articles or other opinions of theologians.
Please note that saying "until the end of the world" and "until the end of time", this article seems to reference the dogmatic constitution Pastor aeternus of the Vatican Council, where the original Latin is "usque ad consummationem saeculi". To convince me, you would have to show that "usque ad consummationem saeculi" in Pastor aeternus does not have the signification found in Mt 24.
Satis Cognitum, Leo XIII: “And, since it was necessary that His divine mission should be perpetuated to the end of time, He took to Himself Disciples, trained by himself, and made them partakers of His own authority. And, when He had invoked upon them from Heaven the Spirit of Truth, He bade them go through the whole world and faithfully preach to all nations, what He had taught and what He had commanded, so that by the profession of His doctrine, and the observance of His laws, the human race might attain to holiness on earth and never-ending happiness in Heaven. (…) But, as we have already said, the Apostolic mission was not destined to die with the Apostles themselves, or to come to an end in the course of time, since it was intended for the people at large and instituted for the salvation of the human race. For Christ commanded His Apostles to preach the "Gospel to every creature, to carry His name to nations and kings, and to be witnesses to him to the ends of the earth." He further promised to assist them in the fulfilment of their high mission, and that, not for a few years or centuries only, but for all time (sed in omne tempus) – ‘even to the consummation of the world’. Upon which St. Jerome says: ‘He who promises to remain with His Disciples to the end of the world declares that they will be forever victorious, and that He will never depart from those who believe in Him’ (In Matt., lib. iv., cap. 28, v. 20).”
Notice, the apostolic mission will last until “the end of time” and will not come to an end “in the course of time.” Also notice that he confirms his statement that the Church will last “for all time,” by quoting Christ’s statement that He will be with the Church “even to the consummation of the world.” This shows that Leo XIII understand “the consummation of the world” to refer to the actual end of time, not a point during the course of time. That’s how the Church and her theologians have always interpreted the phrase.
Leo XIII also teaches that the Church is to remain visible to the end of time:
Satis Cognitum, LeoXIII: “… the Church is so often called in Holy Writ a body, and even the body of Christ – ‘Now you are the body of Christ’ (I Cor. xii., 27) - and precisely because it is a body is the Church visible: (…) those who arbitrarily conjure up and picture to themselves a hidden and invisible Church are in grievous and pernicious error: (…) The connection and union of both elements [i.e., visible and invisible] is as absolutely necessary to the true Church as the intimate union of the soul and body is to human nature. The Church is not something dead: it is the body of Christ endowed with supernatural life. As Christ, the Head and Exemplar, is not wholly in His visible human nature, which Photinians and Nestorians assert, nor wholly in the invisible divine nature, as the Monophysites hold, but is one, from and in both natures, visible and invisible; so the mystical body of Christ is the true Church, only because its visible parts draw life and power from the supernatural gifts and other things whence spring their very nature and essence. But since the Church is such by divine will and constitution, such it must uniformly remain to the end of time (in aeternitate temporum). If it did not, then it would not have been founded as perpetual, and the end set before it would have been limited to some certain place and to some certain period of time (locorum esset temporumque certo spatio defintlus); both of which are contrary to the truth. The union consequently of visible and invisible elements because it harmonizes with the natural order and by God's will belongs to the very essence of the Church, must necessarily remain so long as the Church itself shall endure.”
Pius IX teaches that the Church founded by Christ is a perfect society, with an authoritative teaching body, and that it will always remain as Christ founded it.
Mortalium Animos, Pius XI: “A good number of them [the heretics], for example, deny that the Church of Christ must be visible and apparent, at least to such a degree that it appears as one body of faithful, agreeing in one and the same doctrine under one teaching authority and government; (…) Christ our Lord instituted His Church as a perfect society [a perfect society must be governed by authority], external of its nature and perceptible to the senses, which should carry on in the future the work of the salvation of the human race, under the leadership of one head, with an authority teaching by word of mouth, and by the ministry of the sacraments, the founts of heavenly grace; for which reason He attested by comparison the similarity of the Church to a kingdom, to a house, to a sheepfold, and to a flock. This Church, after being so wonderfully instituted, could not, on the removal by death of its Founder and of the Apostles who were the pioneers in propagating it, be entirely extinguished and cease to be, for to it was given the commandment to lead all men, without distinction of time or place, to eternal salvation: ‘Going therefore, teach ye all nations.’ In the continual carrying out of this task, will any element of strength and efficiency be wanting to the Church, when Christ Himself is perpetually present to it, according to His solemn promise: ‘Behold I am with you all days, even to the consummation of the world?’ It follows then that the Church of Christ not only exists to-day and always, but also exactly the same as it was in the time of the Apostles, unless we were to say, which God forbid, either that Christ our Lord could not effect His purpose, or that He erred when He asserted that the gates of hell should never prevail against it. (…) So, Venerable Brethren, it is clear why this Apostolic See has never allowed its subjects to take part in the assemblies of non-Catholics: for the union of Christians can only be promoted by promoting the return to the one true Church of Christ of those who are separated from it, for in the past they have unhappily left it: To the one true Church of Christ, we say, which is visible to all, and which is to remain, according to the will of its Author, exactly the same as He instituted it.”
If the Church lacks a legitimate hierarchy with the authority to carry out the mission Christ entrusted to His Church, it follows that the Church as Christ founded it no longer exists, which is contrary to the teaching of Pius XI.