Some points are mixed up and needed clarity!
So what is the difference between achiever and failure? Dua’a alone? Right action alone? Or mix of the two? As you said, for non-Momin Dua is out, so it assumes right action alone.
According to the above in the realm of physical world, Dua’a should mean nothing but act of obeying God’s prescribed universal laws. No result without adhering to the laws, since the Kingdom is not governed by magic i.e., no input, expect more output. Otherwise it is hard to live life in magical world, but for some this is such a world if you only know. Where cause and effect is a misguiding mirage.
Obeying and fulfilling His laws are a common denominator to all. It said the differentiating a Momin and non-Momin lies mainly in attributing results to God or to one’s own personal astuteness, respectively. The later may be transgressor of self-aggrandizement, but isn’t that the person was only clever enough to observe the laws? So the forced conventional divide of Momin or Non-Momin on such ground is beside the point.
The real question is, does Dua alone make one knowledgeable, astute and capable to produce results? Going into history, most Dua'as were said at the time when Bahadur Shah Zafar was deposed and exiled. But since God laws were already broken prior to such Dua’as, it was too late to raise hands since the fate was already sealed.
As stated, ‘Dua’a as a ritual plays a significant role in a Momin’s life’. But misunderstanding its clear mechanics of how and by what process, most Momin are reduced to habituated Dua/wazeefa reciters ignoring it’s all other part. Like congregational rain prayers without the thought of changing land topography for its results. But nature has its own way, and what is missing here is the need to learn and obey its command.
Without doing right, Dua’a doesn’t work for Momin or non-Momin alike. Though, it may have its collateral benefit like emotionally or spiritually soothing like balm on the mind. As one spiritual man tutored me if our Dua’a is not answered it is not in vain, it accumulates its benefit in the hereafter. No problem with that.