Actually, the Church does have a memory of this site – the burial cave was found on the grounds of the Dominus Flevit monastery, which is claimed to be the location where Jesus wept over Jerusalem. This is the sort of place that Mary, Martha, and Lazarus might well select for their burial cave – and, if Simon Peter died in Jerusalem, the sort of place he might well be buried.
Which would explain why the ossuaries of Mary, Martha, Lazarus, and Simon bar-Jona were, in fact, found therein.
Considering the eagerness of other ancient sites to be accepted as the burial place of this or that apostle, can you explain why no city but Rome has ever put itself forward to claim Peter?
After the destruction of Jerusalem in AD70 (more than a million killed, sold into slavery, etc?), the place was an absolute mess. How many persons go MIA, how many documents are lost, how many properties are forgotten -- in the middle of a combination genocide/ransacking/slave-raid? The centrality of authority enjoyed by the Jerusalem Church in the days of James’ administration and the Jerusalem Council was no doubt tremendously damaged as a result of the Romans’ laying waste to the entire territory.
But nature abhors a vacuum, and into this gap stepped the nascent Church at Rome – a candidate for central recognition given its placement at the capital of the Empire. It was known, after all, that Paul had come to his martyrdom at Rome; and considering that the great Vaticanus pagan cemetery likely contained quite a few “Peters” (being a common title given to the high prophets and magicians of the pagan mystery religions), it’s hardly surprising that a tradition would develop (a century-and-a-half or two later) as to “Peter’s” burial at Rome. That, however, does not vouch for the authenticity of the tradition.