Could the concept of oxygen pools work?
Yes, but they would be largely transitory, like rain. Most weather mixes the atmosphere quite well, but under some circumstances oxygen pools would form for days or even longer. Perhaps you have oxygen collectors people have to run, and, like rain in the desert, people conserve it. If the atmosphere has pressure, you don't need full pressure suits, you just need little nose cannulas for outside work, and houses which aren't very leaky if you don't want to wear them all the time.
If so, could there be a higher layer atmosphere made up of lighter gasses?
Sure, and we already have that - the Ozone is made up of lighter gases, while down here we have heavier gases. While mixing does occur, the reality is that the concentrations change as you go higher, and you find lighter gases more prevalent, and heavier gases less prevalent.
The weather conditions required for atmospheric pooling might be similar to a phenomena quite well known in cities that are essentially big bowls. It's called an inversion:
It's not going to give you exactly what you want, but a little fudging here and there and you could make this work for a world you have in mind if your audience isn't very picky.
In particularly still areas with no wind, such as in mineshafts, carbon dioxide pooling is a significant issue that has to be actively managed, so this does already happen, it's just not common in large open areas subject to weather patterns.