Use [false-proof] or whatever the tag is to ask about proofs whose flaws are the subject of discussion. This would help prevent the impression of claiming an easy 3-sentence proof of a result that took 40 pages and decades of prior work.
The logic is identical, but leads to a false conclusion, if you replace "prime" by "not divisible by $k$". Given a $(k-1)$-term arithmetic progression [with difference having no factor in common with $k$] of not-divisible-by-k integers, its extension to a $k$ term progression has to have the last term divisible by $k$, even though we were given no divisibility information in advance about that last term. The difference between this and the Green-Tao theorem is that sometimes a string of $(k-1)$ primes can be extended to a string of $k$, and sometimes not. There is a relationship between the first $(k-1)$ items and the $k$th one, but it is more statistical in nature.
The Green-Tao proof, loosely speaking, does use the logic you outlined, but in a statistical, on-average sense. A deterministic guarantee that any $k$ term progression can't have all terms primes (so e.g., if the first (k-1) are prime the last cannot be) would create too much correlation, too much "shared information" about the prime factorization of the different terms. There are theorems in analytic number theory saying that there is at least a certain amount of pseudorandomness in the primes and in the prime factorizations of integers, and knowing in advance that every string of $(k-1)$ primes in progression cannot be extended to $k$ would enforce too high a degree of structure. The reason 40 pages of cutting-edge analytic number theory and Fourier analysis are needed, is to convert the simple heuristics about "information" and "correlation" into provable and well-defined mathematical assertions, which is complicated because the primes themselves are deterministically defined, not a sequence of random numbers to which the probabilistic concepts most naturally apply.