If it a case of "My name is [original pronunciation], but I don't mind if you say it as [adapted pronunciation to accommodate for the other party]", then that is a kindness that the speaker is offering: if I thought I could get close enough to the original without it being tantamount to an insult to his family and language I would try the original, but I would appreciate the offer.
If it is the situation that "The name was originally [Fooian pronunciation], but that contains sounds absent from the language of the place where we now live, so we now pronounce it [adapted version]", then that version is their name, and it would be extraordinarily arrogant to deny people the right to determine what their own name is.
If I see a name like Jorgenson, my default would be to pronounce the J as a Y (but not to try to imitate Danish/Norwegian vowel sounds, because I wouldn't know what I am aiming for, yet alone be confident of emitting them), but once I am told that the carrier of the name prefers to use a J sound, that's the end of the subject.
I think it's a shame that people find it necessary, on migration, to change their name, but I understand it. Maybe it is a kindness to the majority population, maybe it is a path of least resistance, a choice to remove a an obstacle to commerce or opportunity, or simply an admission of defeat and avoidance of embarrassment: it happens.
I wince inwardly when I hear Kinsella or Costelloe said as though they are Italian names, Mahoney as three syllables, or names of Polish origin ending "-cow sky", but that's my fault, not that of the people who use their names in that way. They have the right to choose.
tldr: Original is nice, but not necessarily correct.