[DreamsofPolycarp]

Somewhere along the way, someone decided they didn't like the idea of Peter and his successors as heads of the Church on earth

Well, we might like them if you guys had a clue as to who they were. The first one I see pops up about 379 AD with Pope Damasus.

[MarkBsnr]

http://www.newadvent.org/cathen/12272b.htm has the entire list. I guess your list is a tad abbreviated.

St. Peter (32-67)
St. Linus (67-76)
St. Anacletus (Cletus) (76-88)
St. Clement I (88-97)
St. Evaristus (97-105)
St. Alexander I (105-115)
St. Sixtus I (115-125) — also called Xystus I
St. Telesphorus (125-136)
St. Hyginus (136-140)
St. Pius I (140-155)
St. Anicetus (155-166)
St. Soter (166-175)
St. Eleutherius (175-189)
St. Victor I (189-199)
St. Zephyrinus (199-217)
St. Callistus I (217-22)
St. Urban I (222-30)
St. Pontain (230-35)
St. Anterus (235-36)
St. Fabian (236-50)
St. Cornelius (251-53)
St. Lucius I (253-54)
St. Stephen I (254-257)
St. Sixtus II (257-258)
St. Dionysius (260-268)

[MarkBsnr]

Hmm, as is mine, apparently:

St. Dionysius (260-268)
St. Felix I (269-274)
St. Eutychian (275-283)
St. Caius (283-296) — also called Gaius
St. Marcellinus (296-304)
St. Marcellus I (308-309)
St. Eusebius (309 or 310)
St. Miltiades (311-14)
St. Sylvester I (314-35)
St. Marcus (336)
St. Julius I (337-52)
Liberius (352-66)
St. Damasus I (366-83)