Traditional rock climbers use their ropes and equipment as both a fall-arrest system and a means of ascent. Sport climbers only use their ropes as a fall-arrest system and sometimes and a means of resting on the rock. The same can be done with ice axes - they're a vital means of arresting a fall, regardless of whether you use them to ascend.

If you're determined to climb into the death zone without supplemental oxygen, you could still carry a small cylinder as a bailout option. Hell, if you want to make the climb really challenging, you could carry ten oxygen cylinders and a Piñata.

More to the point, you could just not climb Everest. Other than the altitude, there's nothing particularly interesting about the climb. The Nürburgring was the most interesting and challenging course in the Formula 1 world championship, but the sport abandoned it in 1976 because it couldn't be made safe. The drivers agreed that no race track was worth creating widows and orphans for.