Using a vehicle does not decrease, in any way, the concern about gas going bad. Using the gas which is in the vehicle and replacing it with new would make it a non-event. Mixing the gas itself will do absolutely nothing for it. Using a stabilizer, such as StaBil, will provide you excellent protection if the car is going to sit for a while. (NOTE: I have no affiliation with the StaBil line of products.)
The best way to use fuel stabilizer is as you stated, or, if you put it in when fueling up. This puts the fresh gas in on top of the stabilizer and mixes it as about as good as its going to get. The main concern with letting fuel sit without some kind of stabilizer is most fuels today contain ethanol. Ethanol will absorb moisture, which can cause corrosion within the system (steel fuel lines or steel fuel tank, if equipped). The moisture/ethanol mix can also gunk things up if enough is absorbed.
The other thing which can be done to help is to keep the fuel tank topped off completely. If using fuel stabilizer, utilize the maximum you can. I know with StaBil, it gives you a ratio to use which will allow the fuel to stay good for up to two years. This is not a bad option. Just ensure you put more stabilizer into the tank every time you add more fuel into the tank (enough to continue with the ratio).