It is true that O₂ and CO₂ are generated by lifeforms and in a desert there is less biomass (although microbes are everywhere).

But the reason why there would be no gradients in atmosphere composition is because the atmosphere, specifically the troposphere which extends from the ground up to about 10 km in the arctic and from ground to about 15 km in the tropics is turbulently mixed throughout at all levels and at all times.

If it were not, if it were static then gravity would act and produce a compositionally zoned atmosphere with heavy components such as Xe and CO₂ to be at a lower mean height than other less dense species like H₂O and O₂. This does not happen because the mixing time of the atmosphere is rather short….

If a volcano erupts in the southwest pacific today and pumps a lot of sulfur gasses into the atmosphere THERE, the H₂S and SO₂ gases will be dispersed around the globe in a matter of weeks to months. So, the atmosphere above a desert is constantly being mixed with for example marine air on a timescale of days to weeks if that.

Because of the Earth's climate and landforms (especially mountains and basins) as they are today, it is not possible at this time for the Earth to not have any deserts. However, there were most likely time periods throughout the Earth's 4.56 billion year lifespan where there were no deserts.

The presence or absence of deserts do not really impact the Earth that much - the Earth is constantly changing (just at a rate far too slow for us to notice, except when an earthquake happens or a volcano erupts). If there were no deserts, all of the life (plants and animals) that are adapted to a desert environment would either 1) die, or 2) adapt to a different environment in order to survive.

Deserts form because of the location of mountains and because of the way air circulates around the planet. There have been periods in the Earth's history in which there have been many fewer deserts than today, although the interiors of continents have essentially always been dry.