Metaphorical reference to any visit to a prestigious or desirable site as a pilgrimage? Sure, English does that. In fact, right here on Quora:
Why do people (like to) say “I made a pilgrimage to the Apple Store” as if it were Mecca?
Is it true that all mafiosi must make an annual pilgrimage to Las Vegas?
Pratik Bakshi’s answer to What are some amazing facts about Pokémon GO, both as a game and out in the world?: According to reddit ” Playing Pokémon Go in India is almost a pilgrimage.”
With the anglosphere mainstream being either Protestant or Secular, it’s been quite easy for them to use the word more metaphorically these days.
Remind me Chrysovalanti: is the Greek word you’d use προσκύνημα? Because pilgrimage is certainly one of its meanings, but it applies rather more broadly than pilgrimage: it applies to any veneration of relics or saints, and does not require that you travel far. Of course, the veneration of relics is no more Protestant or Secular than a pilgrimage is.