It’s actually an old debate — in fact, it’s a variation on the Ship of Theseus — but modern day science fiction gives it a new spin. At root it’s a question about exactly where our identity as self-aware conscious beings actually resides. I don’t find it paradoxical so much as intriguing.
As with trees falling in forests, the answer depends on defining key terms. This case depends on exactly what we mean by “dies” and “Kirk” — the latter being the Ship of Theseus.