Because you keep changing water. You kept the levels low. But there is only one fish so you were not creating huge amounts ammonia and by not allowing them to build much things were going on below the level of detection.
You started on June 27th, today is August 14. That is pushing 50 days. A normal cycle should take about 5 weeks or 35 days- give or take
If you followed my suggestion and stopped changing the water so often, then there should have been a change in the ammonia level. If there were not enough ammonia bacs, the level should have risen until there were. If there were close to enough bacs to handle all but .25 ppm of ammonia, then they should have reproduced rapidly enough to make that ammonia go away and stay away. But neither happened in your tank during a week with no water changes.
The one thing I did observe in your reported results is ammonia did rise to .5 ppm and then drop back to the steady .25 ppm early on. So it looks like some movement happened. But consider this. If I asked you to set up a tank and using only fish and other normal tank contents to make the following happen- no matter when the water was tested for ammonia it would always be at.25 ppm. And then to do this for one month. Could you do it? Could you even come up with a theoretical way to do it? As far as I am concerned this would be impossible. But here you are reporting this exact situation in your tank.
My point here is that your numbers are not changing. We know the fish exhales ammonia with every "breath", we know poop and other organic wastes produce ammonia. So if the fish/tank was making ammonia where is it? And the same thing applies to nitrite. If you have ammonia which bacteria are converting, then you must get nitrite. If you must get nitrite where is the reading for it? The only way not to have detected any nitrite that should have been there is if it was being converted to nitrate.
And that gets you to a test kit that is so flakey I don't even bother using it. Your daily water changes minimal stock and lone plant may have combined to make nitrate not being detected either.
And then there is the final consideration. Many dechlors will affect readings. By doing all those every other day water changes, you were dosing more dechlor every time as well. This may have affected your readings.
Lastly, in trying to get info about Aqua One I found this:
Aqua One Ammonia & Chlorine Neutraliser improves aquarium water by removing harmful toxins such as chlorine, chloramines and heavy metals found in tap water as well as reducing levels of toxic ammonia and nitrate produced from fish waste.
I am unfamiliar with this product, perhaps it has something to do with your lack of nitrate readings as well as creating a false low level ammonia reading.
Given all the possible explanations for your readings, the most likely one appears to be that the tank is cycled for the lone betta in it. I could be wrong, but then somebody needs to come up with a better explanation for what went on in your tank the last 7 weeks and for the readings you were and are getting.
