Perhaps the confusion, or your disagreement, is based on timing, or time. To my understanding, to die, the moment of death, is an instant thing, something that happens at a point in time. Whereas the term 'sleep' is used to mean to be in the state of death for a period of time, perhaps for thousands of years before being resurrected. So to 'sleep' means to be in a lifeless state; it doesn't mean 'die'.
So Paul is saying that although many Christians have died, and are 'sleeping' in a dead state, not all Christians will likewise spend time in a lifeless state. Those that are alive when Jesus returns for his body members, his bride, will be changed to have a spirit body, just like those that are resurrected and given a spirit body, and this will happen in a twinkling of an eye. That's fast! The “twinkling of the eye” is the time it takes for light to enter the eye, reach the back of the eye, and be reflected back out. Light travels at 186,000 miles per second, so the twinkle is about one billionth of a second. Blinking is a bit slower (it's a physical movement), but still fast. A blink lasts somewhere between 300 and 400 milliseconds, or considerably less than half a second.
Naturally, the flesh body will no longer exist. Those dead will be given a spirit body - there is no need to bring them to life with a flesh body and then change that flesh body to a spirit body. Those raptured will be instantly changed, which as I understand it means that we will be given a spirit body and our human body will be dissolved away and exist no more - just as Jesus' body seemed to have dematerialised in the tomb. The dissolving of the human body obviously means the death of that body, but we, our souls, will continue to live live forever. As Jesus said, Matthew 10:28 (WEB):
(28) Don’t be afraid of those who kill the body, but are not able to kill the soul. Rather, fear him who is able to destroy both soul and body in Gehenna.