Belief is evidence of belief, not of fact. You should consider Peter Lampe's work, "From Paul to Valentinus: Christians at Rome in the First Two Centuries." Considered probably the most exhaustive work of it's kind, Lampe shows that for the first approximately 160 years, the Christian community at Rome was a diverse mix of house churches with no single presiding bishop. He builds his case on a vast collection of primary source data, which better qualifies as evidence than the faulty history of your Irenaeus, which you quote from his "Against heresies," available here:
http://www.newadvent.org/fathers/0103303.htm
However, out of respect to Irenaeus, it should be pointed out that he is not exactly making out the case for Roman papal primacy. The passage you quoted is part of a larger argument in which he is showing that the Gnostic claim of a hidden tradition passed on by the apostles was a false scheme, because whatever was passed on to the succession of church leadership was done openly, not secretly:
For if the apostles had known hidden mysteries, which they were in the habit of imparting to “the perfect” apart and privily from the rest, they would have delivered them especially to those to whom they were also committing the Churches themselves.
So his argument concerning succession is not that it provides an unbroken line of titular authority from Peter as the key bearer, but that there was a succession of open truth passed forward to each new generation, which truth never had in it anything like the ravings of the Gnostics, as he describes it.
It is also interesting to note he does not latch on to classical Petrine supremacy as the basis for Rome's authority, but mentions both Paul and Peter as organizers and founders of that church, and makes no distinction between them as to authority or supremacy.
Nevertheless, as Lampe's work discovers, Irenaeus may have idealized the history of succession beyond the limits of reality. He was not, after all, an inspired writer, and his writings were subject to error. For example, in another place he gets the age of Jesus wrong because he could not accept that the Pharisees would have described Jesus being less than fifty if he were only thirty, therefore he thinks Jesus must have been forty or older. See here:
http://www.newadvent.org/fathers/0103222.htm
Now no doubt he had good intentions. The Gnostics were making a big deal out of Jesus being thirty because the number thirty corresponded nicely with one of their pet theories about the number of aeons. Irenaeus wanted to take that away from them, but in so doing he introduced unfounded speculation which led to error. That's going to happen to uninspired writers. And that's why the beliefs Irenaeus expressed, while they may provide clues to the truth, must be supplemented with hard facts in evidence that support the claim being made. In that respect, the statements of Irenaeus you have presented fall short of actually proving either Petrine supremacy or succession.
Peace,
SR