It was marketing that was installed on the statute of liberty in 1903, when the U.S. was already fully developed. It doesn’t reflect the original intent at all.
It was written in 1883, as part of fundraising for the pedestal. It might not reflect precisely the "original intent" of the statue, but it's very much in line with all of the other context.
The statute of liberty was from a french admirer of the constitution and abolitionist. It was conceived at a time when Napoleon III had declared himself emperor. The connection to immigration was a completely unrelated glomming-on.
Death of the author. People sailed under the statue to get to Ellis island, it's not a difficult connection to make. The location was known when the poem was presented in 1883, 2 years before the statue arrived in the US and the author volunteered for one of the numerous aid organizations helping jewish immigrants.
The fact that people used it after the fact for marketing an unrelated issue doesn’t have anything to do with the original intent of the statue. There was a lot of ret-conning American history in the late 19th to early 20th century as a result of mass immigration.
Things can come to mean something different from what their funders intend.
It happens all the time, especially with art, language and especially public monuments.
The Statue of Liberty’s connection to Ellis island is undeniable. The national museum of immigration is part of the same monument and run by the same staff.
It’s not ret-conning to say that the Statue of Liberty is indelibly linked - physically and symbolically - to mass migration of working and lower class people. It was the busiest port of entry for more than 60 years, and more than 20 million people entered there. There are uncountable contemporaneous accounts of immigrants viewing their passing the statue as a marker of the end of the voyage, and the beginning of their life in America.
One French guy funded it for one reason. 20 million others saw it as a symbol for something different in their lives.
The full title of the statue is “La Liberté éclairant le monde”—it’s impossible not to see it as a symbol of the ideals of the Enlightenment spreading across the world. That’s the common philosophical ground of both American and French Revolutions, and from there the source of the friendship that the statue represents.
At least some minimal notion of hospitality with respect to migration is part of that Enlightenment. (Kant’s Perpetual Peace is emphatic about this; Derrida annotates the relevant section with fresh eyes in Hospitality vol. 1, the first lecture and ff.)
That said, I also agree with you that symbols are not fully formed at birth and it is not the case that what they represent never changes at all in the course of their history.
