It is usual for PhD students to have tuition and a small stipend for their living expenses paid for by government. The government knows perfectly well it needs some supply of researchers, the old ones eventually die so you need to train new ones and a PhD course is how you do that.

A particular university Computer Science department might have say, six slots this year for PhD students. It needs not only the money (which you could if you're wealthy just pay for yourself) but also experienced staff willing to supervise these students which may be a more limiting factor. If it has eight capable applicants versus six slots then yes, you might get refused, but it's rare for there to be a huge imbalance between the number of applicants and available slots such that you'd need to be a "genius".

If you have a good (say first or upper second honours) undergraduate degree in the same or closely related area, and you can find a PhD topic and a doctoral supervisor, in Computer Science they can probably make the money happen.

You will not be wealthy (the stipend is small), but you should be too busy to notice and you're hanging out with other students who haven't got any money anyway.