The main reason why people say you MUST do 100% water changes on ANY SIZED UNCYCLED TANK is because of the compounding ammonia build up.
For science sake, lets say you have a robotic betta that will not die from ammonia poisoning and it lets out 0.5G of ammonia a day, living in an uncycled 1 G tank. FOR SCIENCE SAKE.
First day: Tank is 0.5G ammonia, 0.5G water
100% water change --> 1G water, 0G ammonia
50 % water change --> 0.75G water, 0.25G ammonia
Second day: Add 0.5G more ammonia
100% tank from yesterday: Now has 0.5G ammonia, 0.5G water, another 100% change will bring it back to 0G ammonia, 1G water
50% tank from yesterday: Now has 0.75G ammonia, 0.25 water, another 50% change will bring it to 0.37G ammonia, 0.62G water
Third day: Add 0.5 G water
100% tank from yesterday: Now has 0.5G ammonia, 0.5G water, another 100% change will bring it back to 0G ammonia, 1G water (THIS IS CONSIDERED A CONSTANT DUE TO 100% CHANGES)
50% tank from yesterday: Now has 0.87G ammonia, 0.13 water.
Notice how regardless of how many 50% water changes you do, without bacteria to remove it, or removing it by your own hand, it will just continue to grow horrendously concentrated. Considering it is in ppm (parts per million), and how sensitive fish are to ammonia, your 50% water changes in an uncycled tank will quickly become ineffective.
However, as vaygirl said, sometimes it is simply not your fault, and it was probably something out of your control from the tap. This is why I always make it a good habit to test my water before a water change and after a water change before I put my fish back in (in both uncycled and cycled tanks). Just in case your pH is suddenly super different or your water company decided to dump in an extra stick of chlorine or they forgot some water softener, etc.
As for your question about getting a new betta, this is how I acclimate them out of the blue water and into their new home:
1. Get a QT tank, fill it up halfway-ish with TREATED water, let it sit for a few minutes (if it has a heater, even better)
2. Float new betta in his/her for 20-30 minutes
3. Use his/her cup to scoop very little water, let them float again for another 5-10 minutes.
4. Gently pour half of the water in the cup into a bucket (not the sink, you don't want to accidentally lose your fish DX
5. Float for 5 minutes, then scoop a little more water, 5 minutes, more water, 5 minutes. (Empty water as necessary into the bucket)
6. After feeling that the blue water is MOSTLY transparent, regular water colour, I tip the cup so that they have enough room to swim out. If they are ready the will swim out on their own, usually (in my experience at least)
7. After they are swimming around in the QT, I do 50% water changes every 10 minutes until the water is definitely clear, then I float them in their home for 20 minutes, scoop once or twice then release.
Hasn't failed me so far >_> *knocks on wood*