What is the biggest reason why the Mount Everest climbing window each May and September is so short and often less than a week?

It's all about wind and temperature.

It is only when the winds die down in May and again for a short period in September, that we have a so called 'Summit Window', when conditions are safe enough for climbers to try and reach the summit.

https://explorerspassage.com/chronicles/facts-mount-everest/

Lower wind speeds in May and warmer temperatures explain the relatively short climbing window. Wind chill and low temperatures combined with high elevation make the summit of Everest one of the least hospitable places on the planet during the winter.

https://www.himalayanwonders.com/blog/mount-everest-base-camp-weather-climate.html

At almost 9km above sea level, the summit of Everest is already a very cold and windy place. Being in the Northern Hemisphere, Everest experiences its winter over December to March. While it can be summitted in winter, it's not an ideal time due to the extreme cold.

From June to August, monsoon season hits Nepal, causing significant snowfall and storms which are very dangerous on an Everest expedition.

Thus the best window to summit is from mid-April to late May where conditions are warming up but the monsoon has not yet arrived. The weather also tends to be more stable with more sunny days and fewer storms. Another window exists in September to November after the monsoon has ended but before the cold winter sets in a again, however this is much less popular as the increased snow cover does increase the difficulty of climbing.

As for why the suitable period to reach the summit can be less than one week, we're actually taking about 'summit windows' rather than climbing season. Even during the climbing season, most days are not suitable for summit attempts due to high wind or other factors. Expeditions to climb Everest take over a month to allow for adequate acclimatisation. Once acclimatised, teams will wait at Base Camp for a period of good weather to be forecast, which may be less than a week. They will depart from Base Camp and aim to reach the summit within a few days during this window.

Due to the limited number of summit windows, the schedules of most teams will align, leading to the vast majority of successful summits occurring within a very short period each year. This causes big traffic jams on the summit push, which have been increasingly causing concern due to the danger they pose to all climbers.