A Priestless Church was good enough for the Apostles. What is wrong with a Priestless Church?
Incorrect. Jesus was of the priesthood of Melchisedech as Psalm 109 affirms and the word priest first appears in Scripture in Genesis 14:18 in reference to Melchisedech. Jesus made the Apostles the first priests in His Church.
The Apostles were, each and every one of them priests and bishops. The earliest fragments of the liturgy of the Mass date to approximately 120 AD, when disciples of the Apostles were still in leadership such as St. Ignatius of Antioch. Those fragments are probably older than the earliest existing copies of the books of the New Testament.
It would be more historically accurate to say that the early Christian (i.e Catholic) Church existed without copies or even originals of substantial portions of the New Testament since those portions had yet to be written than to suggest or claim or declaim that it existed without priests.
For example, it is manifestly obvious that the early Christian Church and some of its activities are described in the Acts of the Apostles as history and not as prophecy. Obviously, St. Paul was writing after the facts described. St. Stephen was martyred by Saul and his non-Christian associates before the former Saul could write about the incident as Paul. The Apocalypse or Revelations seems to have been written very late in the first century and after the deaths of most of the twelve Apostles and many of their contemporary priests.
There is a historical continuum that ought to be observed by those who argue in good faith (small f). On a matter of such importance as the Christian Faith, people living in 100 AD had a good idea of what happened in 50 AD and passed it on orally and in writing and discussed such matters as earnestly as we do today. Substitute freely each subsequent 50 year period and cumulate because, just as we care what happened in, say, the 2nd century as to Christianity and Christians, so have our predecessors. In this sense, it is more than a little pretentious to assume that what is not in Scripture did not occur in spite of any and all non-denominational evidence. It would fall into the same category as the calculations of Bishop Usher of the soi disant and misnamed (Anglican) Church of Ireland that the world was created approximately 4,000 B.C because of his wrinkle on the ages and intersections of the lives of the old Testament patriarchs or the idea that dinosaur fossils are a cosmic joke because they could not have existed since the Scriptures fail to mention them.
The answer to your question is: Without those specific priests who are validly consecrated as bishops, there can be no Apostolic Succession. Without Apostolic Succession, there is no authority to ordain priests. Without priests, there will be none of the following sacraments: the Holy Eucharist (and the Real Presence of Jesus Christ: Body and Blood, Soul and Divinity therein), Confirmation, Penance, Holy Orders, Extreme Unction. In such an event, the gates of hell will have prevailed which is impossible because it is guaranteed in the Peter passage of Matthew that they will not and the Scriptures tell no lies.