A good chunk of what you wrote has certainly already been formalized and proven.

I love the visual of cutting a cake and seeing how much is left as a portion of the whole. This line of thinking leads one to the idea of "natural density", in roughly the same terms you gave, but with the help of limits.

But natural density is kind of broad and you need lots of hedges to squeeze out something useful when it comes to slicing up slices. Luckily for you, you have noticed one major hedge: the candidates form a cyclical pattern.

And this is where number theory (where you should start if you are interested in primes) comes in. We call a set of numbers a "reduced residue system modulo n" *exactly* in the situation you have described: it's what's left over in one cycle of length n after you take out all the stuff that isn't coprime to n. So from 7 to 12, the set {7,11} forms a reduced residue system modulo 6.

A residue class is what you get when you consider all the "equivalent" candidates (for example, in your k=2 example, 7 and 13). The nicest thing about residue classes is that they form arithmetic progressions.

And that's the ingredient that you'll need to make the density/slicing argument work, essentially that the "slice" is evenly distributed among the whole cake and won't be intefered with by past slices. Your infinitude of primes "proof" is good.

Now I'm no expert in number theory or anything like that, but I think where your idea starts to fall apart is that when you slice out primes, each prime affects only its own kind, but with twin primes the removal of one prime can knock out up to two others: for example 13-15-17 is *two* sets of twin prime candidates, but knocking out 15 knocks out both (13,15) and (15,17) from candidacy. (I know the example isn't perfect, cuz 11-13 still holds, but hopefully you get the idea).