How can I build my own water aeration system for radon mitigation?
There are a variety of risks, including the risk that the system will fail to work as intended. You either punt those to let someone else design the system, or you take them on yourself when you design it. 

To look at just one that you've thought of: Not growing bacteria in your water system is pretty much a matter of having clean water to start with and good sanitation when you work on the system. If your well is not contaminated, then sanitizing your water system whenever it's opened up for work should mean your system remains free of bacteria, if you don't introduce them. So, you probably need a filter (capable of excluding bacteria) on the air you'll be blowing through the packed media filter, for instance. Often, when you add up all the parts you need to do the job right, prebuilt systems don't look as expensive as they do when you haven't looked at the problem in depth; but that's not obvious until you do look into the details of the problem and solving it, reliably, completely, and without adding other problems.

If the air is not adequately filtered, you may need to add in a chlorination or other disinfection system after the filter to kill bacteria introduced in the filter.

Also - a packed media filter should reveal a few things to you in a search - it's more spraying the water over a physical medium that allows it to drip a lot than blowing bubbles. You need the booster pump because you lose all water pressure at this stage, so you pump from your well to the radon filter, and from the radon filter to your house. The thin layer of water on the media is more amenable to removing/exchanging gas than bubbles or droplets. The same type of system is often seen in waste treatment where a great deal of oxygen exchange is needed.

This from PSU extension has a pretty good discussion/description.

Here is a bit more on packed columns - you are interested in the process type called stripping or desorption. 

This one is about removing VOCs and suffers a bit in translation but has good illustrations and removal of VOCs is the same basic process as removing radon.