Metaphorical reference to any visit to a prestigious or desirable site as a pilgrimage? Sure, English does that. In fact, right here on Quora:
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.