I believe that Christ himself is the head of the Church.
No one of faith disputes that. And Jesus set up His hierarchy for His Church

R:
I believe that St. Peter held a special place or preeminence among the Apostles.
Hold that thought
R:
Whether it was Christ’s divine mandate that that role be passed on to those who would succeed him at Rome, I do not know.
How can you believe the former and not the latter, given that the latter, is Peter’s see?
R:
I accept that the ancient Church-both East and West-believed that the bishop of Rome in fact does hold a role of preeminence among the bishops of the Church.
Then that answers your previous question. It came from Jesus. Speaking of early Easterners, Irenaeus from Smyrna in the East, 1 man away from an apostle, who was made Bp of Lyon France in the West by the pope, wrote that it is apostolic teaching that the office of government in the Church, was passed down in succession from the apostles. Therefore, Peter’s office likewise went to his successors.
Bk 3
Chapter 3 , v 1- 3
(all emphasis mine) the link gives full context
" …[snip] For they were desirous that these men should be very perfect and blameless in all things, whom also they were leaving behind as their successors, delivering up their own place of government to
these men; which men, if they discharged their functions honestly, would be a great boon [to the Church], but if they should fall away, the direst calamity.
Note: Irenaeus names who “these men” are, in v 3 & 4 that follow. ]
2. Since, however, it would be very tedious, in such a volume as this, to reckon up the successions of all the Churches, we do put to confusion all those who, in whatever manner, whether by an evil self-pleasing, by vainglory, or by blindness and perverse opinion, assemble in unauthorized meetings; [we do this, I say,] by indicating that tradition derived from the apostles, of the very great, the very ancient, and universally known Church founded and organized at Rome by the two most glorious apostles, Peter and Paul; as also [by pointing out] the faith preached to men, which comes down to our time by means of the successions of the bishops. For it is a matter of necessity that every Church should agree with this Church, on account of its preeminent authority, that is, the faithful everywhere, inasmuch as the tradition has been preserved continuously by those [faithful men] who exist everywhere.
Irenaeus lists “These men” 12 Bp’s of Rome from Peter down to his day].
3. The blessed apostles, then, having founded and built up the Church, committed into the hands of **Linus **the office of the episcopate. Of this Linus, Paul makes mention in the Epistles to Timothy. To him succeeded
Anacletus; and after him, in the third place from the apostles, **Clement **was allotted the bishopric. This man, as he had seen the blessed apostles, and had been conversant with them, might be said to have the preaching of the apostles still echoing [in his ears], and their traditions before his eyes. Nor was he alone [in this], for there were many still remaining who had received instructions from the apostles. In the time of this Clement, no small dissension having occurred among the brethren at Corinth, the Church in Rome dispatched a most powerful letter to the Corinthians, exhorting them to peace, …
[snip for space]
To this Clement there succeeded
Evaristus. Alexander followed Evaristus; then, sixth from the apostles, **Sixtus **was appointed; after him,
Telephorus, who was gloriously martyred; then
Hyginus; after him,
Pius; then after him,
Anicetus. **Soter **having succeeded Anicetus, **Eleutherius **does now, in the twelfth place from the apostles, hold the inheritance of the episcopate. In this order, and by this succession, the ecclesiastical tradition from the apostles, and the preaching of the truth, have come down to us. And this is most abundant proof that there is one and the same vivifying faith, which has been preserved in the Church from the apostles until now, and handed down in truth…[snip]"
R:
I am very skeptical about claims that the ancient universally believed the role of the bishop of Rome extended to universal ordinary jurisdiction and papal infallibility, as understood since Vatican I.
The claims are not mere claims. The evidence is there to prove the Catholic position