It is one of the great wonders of the natural world, and it is a place that many people hold to be very special. It is, in a way, an almost "sacred" place, in a weird secular sense. It is not just any other mountain. There's a reason people are queueing up to see the summit.
Maybe I'm a crazy person, but I think that we have a moral obligation to protect the great natural wonders of the world. You can visit them, sure, but only if you do so in a responsible way that ensures their preservation. I do not see how that's possible with Everest any more. History has shown that there is no way to guarantee that you will not be forced (by circumstance alone) to leave trash on the mountain. That trash, by the way, will become essentially unrecoverable: it will likely be there for decades or centuries.
Same thing with your human body: if you want to risk your life, that's fine. If you want to drag others into your idiotic quest to pretend to be Edmund Hillary, that is also fine (as long as they do so full consent and knowing the risks). What is not fine is that if you do die, your body will be essentially unrecoverable, and it will also litter the mountain for decades or centuries.
If you care AT ALL about preserving Everest (and I suspect most people do), I claim there's no morally permissible way to climb this mountain. Leave it alone. Go climb something where you can make sure you can take your refuse with you, and where your dead corpse can be recovered.