The root zone gets O2 in several ways:
1) O2 dissolved in the nutrient solution prior to use is carried into the root zone
2) As you feed, the turbulence of the flow of nutrient solution around media particles makes more O2 dissolve into the solution. (this is where watering fast vs slow comes in)
3) As the media drains and dries out, air penetrates into the spaces between the particles of the media This creates a wet media surface in contact with air and O2 can dissolve into the solution that remains.
It's all about the contact between air and liquid. The best way to ensure your root zone has sufficient O2 is to start with a media that does not compact to much in the container and had good drainage (Peat, Coco etc, are all good) and to feed using solution that has O2.
Everyone's water source is different, many household faucets have little aerators that add bubbles to the stream as it comes out but some people fill their tanks from a hose. Here are a few ways that you can make sure your starting nutrient mix has as much O2 as possible (in order of my personal preference).
Best- Add a small air pump and bubble stone and let it run for 15-30 min before you feed and if you have extra solution you can leave the air pump on and keep it for the next feeding. For a small batch of nutrient a 10-15$ pump will be fine you don't need anything fancy but the deeper your res tank the more power you need to get air to the bottom.
Good- Use a small (SMALL) water pump at the bottom of the res tank, pointed up. This will make the surface turbulent like a pot of boiling water help O2 dissolve.
Basic- Put your solution into a bottle and shake it gently for a few min before feeding.
On the topic of run off, if the plants soak up the solution from the trays then is not really run off. Its showing the entire media is saturated but everything still remains in the pot at the end of the day. Having 10% run off is not needed every time in peat media but its not a bad idea to do it every few feedings.