AFAIK I know it is a "core activity" in one Abrahamic religion (Islam)—more so than in any other major religion.

A lot of Christians I know have done some sort of pilgrimage. Quite a few Buddhists too (most of those I know if you count short and easy trips).

I agree it is very common, but I cannot see a correlation between how ancient a religion is and how important pilgrimage is within in it.

Before the destruction of the Second Temple, pilgrimage was an important part of Judaism as well.

There are quite a few Muslim peoples who don’t recognize the hajj to Mecca as an obligation, such as Nizari Ismailis and Alevi-Bektashis. Even in the Balkans or West Africa among Muslims who ostensibly subscribe to a more “orthodox” Islam, though they might see the hajj as a nice thing, consciousness of it as an obligation may be low.

There's a big difference in Christianity between Protestants and, for a lack of a better word, the Apostolic Churches (Roman, Greek, Oriental).

The later have a very rich culture of pilgrimages.