The actual influence the papacy has on the way everyday Catholics actually think about their religion is not that large. For instance, arguably, most Catholics are actually heretical (well, not in the sense of 'in heresy from the Orthodox Church', which they obviously are, but in the sense of 'in heresy from their own Church') as far as things like basic salvation go. This has a lot to do with the way Catholicism is taught and popularized by local parish priests and things like doctrinal instruction in confirmation.However, the Vatican does have a disproportionate influence on Catholic theology, but only because the Catholic church is structured in a way that it should - Islam doesn't have a structure in the same way at all, doesn't have a head, doesn't have one recognised body to interpret the scriptures, and so on. Most Catholics agree that the Pope knows what he's doing most of the time; there's nobody that can really occupy that position for Muslims.
You're also going about the argument the wrong way. If somebody thinks that the presence of Makkah and Madinah in Saudi Arabia means that Saudi interpretations of Islam have disproportionate influence on non-Saudi Muslims, that person obviously would think that the Papacy has a disproportionate influence on Catholics worldwide. Saying that "the comparison's invalid because the Papacy actually has a hierarchy" just strengthens my case and proves you haven't been reading.