Properly oxygenated water is definitely a good thing.
With 1 gallon, just shake it for 30 seconds, and it'll have plenty of O2. There's no reason to leave 1 gallon stagnating over time, so no reason to put an airstone in 1 gallon of water. Just fill as you need it. The rushing water out of the tap will oxygenate it too.
Diffusion alone is a bit slow, but churning the surface is significantly more effective at adding O2 to water than air bubbles of any size floating to the top.
Also, water temp and a properly constituted medium with good drainage and aeration characteristics are equally important. Any over-saturation of O2 is just going to quickly gas off once its in the pot, so don't overthink this.
LArger volume reservoirs should use a powerhead, not an airstone. Maybe anything over 20-30 gallons? I'd look at when fish aquariums start recommending them.. based on same concepts.
there's a video on youtube with some fish struggling to aspirate and congregating near surface of some small man-made water feature. The guy turns on the little fountain or sprays a hose on surface and within seconds you can see the fish's stress alleviated and no longer congregating for limited O2 near surface. that shows you how fast O2 can rise with some agitation. Definitely doesn't take 1-2 hours.