It's not. Jim1 and Jim2 are both 100% Jim, and both are equally "the real Jim" in every way that matters. That Jim0 was vaporized at the moment of creation doesn't change the properties of Jim1 and Jim2, nor would it change the properties of, say, Jim3 and Jim4 when they are created a week later from a backup copy leftover from the original scan.
Not quite. From the POV of Jim0, he no longer has a POV because he is dead.
Sucks for Jim0, but irrelevant for Jim1 and Jim2; the pov of a being or beings accurately called "jim" has been preserved. One iteration of Jim has been suspended, but two others still exist.
I repeat: all three Jims are distinct individuals,
but at the instant of their creation, they are identical to Jim0. For all intents and purposes they are indeed Jim, but different iterations of Jim still have separate existences despite the fact that they are Jim.
Or let's put this another way to make it easier to understand. You have one Human. You then copy this human's DNA and scan it into a computer. You then re-sequence the DNA of a fertilized gerbil egg and place it in a Borg maturation chamber and VOILA! You have a human, identical to the human you cloned. This clone, though created artificially, is still a human in every meaningful sense. Why? Because it has a full human genotype and a full human phenotype: all the characteristics that make a meaningful definition of "human."
In Jim's case, what you have copied is basically a "memotype," that is the full set of mental/psychological/memetic data that makes "Jim" the person he is. As "Jim" evolves in his life, that basic structure changes slightly, just as a cloned human's offspring will have different traits of the original (even if they both marry the same woman; genetic roll of the dice).
Try wrapping your brain around this: at the moment, human identity is singular; that is to say, there is usually only one of each of us. There is no logical reason why this MUST be the case; just because we're used to the idea that there is only one "Jim" doesn't mean that "Jim" may not also describe an entire CLASS of individuals all incredibly similar to Jim. Consider, for example, that while there is only one Enterprise, there are several Constitution class ships in the fleet.
Whether Jim1 is just as real as Jim0 as far as everyone else is concerned doesn't change the fact the Jim0 now experiences death and nothingness. So from Jim0's POV...
Jim0 no longer has a pov if he is dead, so that's a non-starter. OTOH, if Jim0 is still alive, it remains the fact that his pov is just as valid as Jim1s, because they are both "Jim." BECAUSE they are both Jim, both of them believe they have just experienced the act of scanning and duplication, and both of them are literally correct. What differs is their point of view AFTER duplication; then and only then do their povs diverge.
Jim1 is NOT real in the way that matters the most to him, which is for Jim0 to continue being Jim.
But the same goes in reverse: Jim1 ALSO continues being Jim, because he has his own pov in which Jim0--who he is looking at--isn't as "real" as he is. Only the self is completely real from any particular point of view, and unless Jim1 is a holographic simulation on a disk somewhere, both povs are equally valid.
And if people understood that they aren't actually being transported but "cloned" and killed in the process, I imagine the vast majority of people wouldn't use it.
Transporters don't work that way, so it's a non-issue. OTOH, if the transporter really was a cloning device, you better believe it would be used ALOT of different ways, particularly in espionage and intelligence gathering, not to mention--maybe in the most important case--the replication and mass production of Data as per Commander Maddox's wet dreams. Transporter-as-cloning device just isn't consistent with how they're used, and more importantly, how they're NOT used; such a device would render death fundamentally obsolete and would have extremely disruptive implications for the very notion of identity and individuality that would make episodes like "The Masterpiece Society" fatuous at best.