Hydrogen alpha emission is generally stronger than hydrogen beta, and occurs at a more conveniently observable wavelength. Hydrogen beta is bluer than H-alpha, and suffers more atmospheric attenuation as well as being a weaker line.
Hydrogen alpha isn’t actually the strongest line that ionised hydrogen atoms produce – that’s Lyman-alpha. But it’s in the ultraviolet and can’t be observed from the ground, so once again practicalities get in the way.
The ratios of all these lines are interesting, but if you’re going to measure one single line, H-alpha is where you’re going to get the strongest detection.
The CNO cycle is a mechanism which produces energy in stars which are somewhat more massive than the Sun. It requires a temperature of about 17 million kelvin, whereas the Sun’s core is only 15 million K. Even in stars which are powered by the CNO cycle, it’s restricted to the very centre of the star. The visible photosphere will be only a few thousand K and what you’ll see is mostly H and He.