The download option admittedly brings with it a VERY uncomfortable world-view with it: that all intelligence, sapience, etc., is nothing special, nothing magical, that it is merely a side-effect of properly wired circuitry. It also admittedly removes a lot of the various religious and pseudo-religious / spiritual aspects of things. These are a security blanket for most of humanity, and therefore, by extension, and of our creations, fictional and real alike.
You ARE correct, though: the original "substrate" still dies, while the program instead is run in a different medium. I admit, yes, it's a mechanistic as heck opinion and worldview, but such is. It works on certain assumptions, none of which can be currently proven for or against without resorting to religion and spirituality on one side, and their defeat or removal from things.
I shall avoid getting into the social, political, rights, and other ramifications that occur when both the original AND a downloaded copy are in simultaneous existence. That's a thorny ball of wax best left to the bioethics experts.
On the mention of immortality... The odd thing is, we've already, IRL, conclusively and repeatedly engineered a "immortal" fruit fly. The technology is, barring corporate and governmental interference, probably, at a (not too educated) guess no more than a century away. Thus, I can't see the tech being out of the grasp of a sufficiently advanced race. However, as a caveat, there are MARKED social and economic issues that flag themselves very quickly as critical once a species reaches immortality, whether it be via the traditional methods (engineering themselves not to age), serial immortality (repeated downloads to a young, virile cloned body, there to live however long medical technology can keep you alive), or other methods, some of which do not include bodies, and others involving multiple bodies housing the same sentience. These issues can splinted a society, leaving fragments spinning off into the socioeconomic void.
On a personal level, rather than anything intellectual, I think some of this "incurable" issue has its' roots in the religious/spiritual belief that immortality isn't the provenance of we mere mortals, and that it is hubris to try to take such from the hands of God/the gods/etc. While some might think me insane, or even equally filled with hubris, I could not disagree more strongly. I believe we can and shall one day use this as commonplace.
Whoa. I'm unusually verbose today. Must be me being sick. well, there you have my opinion on the matter.