As others have said, using supplemental oxygen increases the partial pressure of oxygen in a climber’s lungs, allowing them to breathe normally despite the lower pressure. There is obviously a theoretical limit to the altitude you could reach using supplemental oxygen alone and beyond which you would need some sort of pressurization scheme, but I believe that altitude is higher than the summit of Everest*. Currently, climbers limit the degree to which they use supplemental oxygen they use because of the difficulty of getting it up to high altitudes, not because they are at the limits of where supplemental oxygen would be beneficial. If you could get more oxygen up to the high camps, climbers could use more supplemental oxygen which would improve their performance, no pressure suit required.
Furthermore, regarding using pressurized masks, I would point out that much of World War 2’s air battles were fought in unpressurized aircraft with aircrews who went from sea level to Everest-level and higher altitudes in a matter of hours wearing only pressurized face masks. For example, the oxygen system on the B-17 would pressurize and supplement such that the oxygen content breathed in by aircrews was equivalent to around 10,000 feet and was rated up to 40,000 feet. You’re right that you eventually need a pressurized suit or vessel, but it’s at a point much higher than merely Himalayan altitudes.
*In early high-altitude balloon experiments it was found that balloonists could reach 35,000 feet by breathing unpressurized pure oxygen. So that could be the limit of what altitude can be achieved by supplemental oxygen alone, although I don’t know if “reached” means the balloonists could survive comfortably for any length of time or merely that they didn’t die.