Mt Everest didnt suddenly grow because it was 'discovered' and named by westerners. Its been tallest for many millenia. However it is believed that the Incas may have climbed as high as 23,000 feet in the Andes mountains, specifically Aconcagua and Llullaillaco peaks. In 1905 climbers reached almost 24,000 feet in Ghurla Mhandata and later Tisule peak at 23,360 feet. Befor Everest the highest asscent was Anapurna at 26,300 feet. K2 is the second highest peak in the world.
Mt Everest has always been the tallest mountain on earth; it is just that it had not always been known. Far from this, Mount Kangchenjunga in Nepal was initially thought to be the tallest mountain on earth, and them Mt Everest happened.