The Wikipedia article about Peter has a section about his connections to Rome. The references in that section should be helpful.

Also, what specifically is the claim you are asking about? That Peter was the first pope? That he was the first bishop of Rome? That he actually went to Rome? There are ways he could still be considered the first pope/bishop of Rome even if he never physically went there.

@RobWatts The question is, obviously, not asking whether/why Peter was considered a pope or a bishop. The claim that's being questioned is that Peter had anything to do with Rome, e.g. that he travelled there, lived there, and was martyred and buried there.

@ChrisW That's not how the question reads - the first line says the Catholic church claim St Peter as the first pope/first bishop of Rome, and the last line refers back to this claim. By itself, this does not require Peter to have ever been to Rome.

@RobWatts The question isn't whether he was a "bishop", it's whether he was a bishop "of Rome". The other sentences in the question ("A church already existed in Rome", "Peter's primary responsibility was for Jewish people, not the Romans", "no evidence that Peter ever went to Rome") make it clear that the question is about Peter's alleged connection with Rome.

catholic.com/tracts/was-peter-in-rome claims that "a key premise of their argument is the assertion that Peter was never in Rome" ... I assume that this (that Peter was physically in Rome) is the claim being questioned here.

I don't think I've been doing a good job of explaining myself. I'm trying to point out a discrepancy between being a "bishop of Rome" and having actually been in Rome

How does being a "bishop of Rome" automatically require that Peter have gone to Rome at some point?

Isn't it possible that he could be viewed as the first bishop of Rome in a metaphorical sense, rather than a literal sense?

It is metaphorically possible, yes: for example, Queen Victoria was "Empress of India" without actually visiting India. But, Catholics allege that Peter was physically in Rome, see e.g. en.wikipedia.org/wiki/Saint_Peter's_tomb and IMO that's the claim that's being questioned.

That's what it looks like to me, too, but I was wanting to have the OP confirm that. If that is the case, then the question should be changed to make that more clear.

The one hyperlink in the question is to Otto Zwierlein's interview, titled "Has St. Peter ever been in Rome?".