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“They want all to actually be catholic priests.”

According to your catechism, they already are:

“1268 The baptized have become “living stones” to be “built into a spiritual house, to be a holy priesthood.”74 By Baptism they share in the priesthood of Christ, in his prophetic and royal mission. They are “a chosen race, a royal priesthood, a holy nation, God’s own people, that [they] may declare the wonderful deeds of him who called [them] out of darkness into his marvelous light.”75 Baptism gives a share in the common priesthood of all believers.”

http://www.vatican.va/archive/ccc_css/archive/catechism/p2s2c1a1.htm

Though, this seems a fairly meaningless concession, since the Catholic church doesn’t seem to think members of this priesthood are able to perform priestly duties.

Dear Boogieman, there's a priesthood of the laity, and then there's the ordained ministerial priesthood. Two different things. I'm sure St. Paul wouldn't have thought it was OK for people claiming "We're all preists" to have shoved out Timothy and Titus, who were appointed to the task by Paul and ordained by the laying on of hands.

It is evident that the Apostles had authority over the Church, and that authority was also given by the Apostles to bishops, priests, and deacons:

"We beg you, brothers, respect those among you whose task it is to exercise authority in the Lord and admonish you; esteem them with the greatest love because of their work" (1 Thess 5:12-13).

These leaders were also given the authority to teach. Timothy and Titus were bishops who were repeatedly instructed to teach. The bishops, who ruled the Church and had a position of authority, were instructed to teach:

"Let the elders who rule well be considered worthy of double honor, especially those who labor in preaching and teaching" (1 Tim 5:17).

It was the role of the bishop to teach the true gospel and defend it against false teachings:

"In his teaching [the bishop] must hold fast to the authentic message so he will be able to encourage men to follow sound doctrine and refute those who contradict it" (Tit. 1:9);
"Exhort and rebuke with all authority" (Tit. 2:5,15);
"Instruct . . . command and teach . . . give attention to reading, to exhortation, to doctrine" (1 Tim. 4:6,11,13);
"Convince, rebuke, and exhort, with teaching" (2 Tim. 4:2);
"Teach and exhort" (1 Tim. 4:2).

They received these truths and this authority from the Apostles:

"Do not neglect the gift that is in you, which was given to you by prophesy with the laying on of hands of the presbyterate" (1 Tim. 4:14);
"Hold fast the pattern of sound words which you have heard from me" (2 Tim. 1:13);
"Guard the rich deposit of faith" (2 Tim. 1:14).

Not everyone was authorized to be a teacher. There were only "some teachers" (1 Cor. 12:28-29; Eph. 4:11), who were the bishops and elders (priests) of the Church.