b. For Green-Tao and related results the following as some 'warning': The sixty-first Putnam competion (2000) had the following question (paraphrasing): the values the polynomial $Q=X^2+Y^2$ takes on $\mathbb{Z}^2$, contains inifinitely many triples of consecutive integers. This is not hard to show. Yet, then in the book "The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary" by Kedlaya, Poonen, Vakil there is mentioned the related question (p.279), (then) stated as open problem (again prarphrasing):

Yet, I will try to demonstrate by an analogy, somewhat close to the problem at hand, at least in my opinion, what is about theintuition.

Recall that Lagrange's Four Squares Theorem asserts that every nonnegative number is the sum of four squares of integers. Or in other words, for $L= X_1^2 + X_2^2 + X_3^2 + X_4^2$ one has $L(\mathbb{Z}^4)=\mathbb{N}$ (where we choose the convention that $\mathbb{N}$ includes $0$).

Now, this is on the one hand an interesting result in its own right, and on the other hand the fact that one can characterize the nonnegative integers among all integers in such a way/by such a formula is sometimes also used. In fact, it is used in considerations close to the one at hand see its mention in the context of Hilbert's tenth problem.

So this is fine and interesting. What however does not seem like a very feasible idea is to try to understand the nonnegative integers better by analysing the polynomial $X_1^2 + X_2^2 + X_3^2 + X_4^2$.

And, to some extent the situation for the primes and this polynomial seems comparable. Yes, one has this characterization of primes, this is very interesting but not so interesting to understand the primes (in the sense, say, some analytic number theorist would like to understand them, frequency, gaps, etc.): the description is not very convenient to work with, the primes do not come out in a systematice way, there are various polynomials having the sam property (so why this and not another), the fact that such a polynomial exists is nothing very specific to the primes but also true for all recursvely enumerable sets, and so on.

b. For Green-Tao and related results the following as some 'warning': The sixty-first Putnam competition (2000) had the following question (paraphrasing): the values the polynomial $Q=X^2+Y^2$ takes on $\mathbb{Z}^2$, contains infinitely many triples of consecutive integers. This is not hard to show. Yet, then in the book "The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary" by Kedlaya, Poonen, Vakil there is mentioned the related question (p.279), (then) stated as open problem (again paraphrasing):

Yet, I will try to demonstrate by an analogy, somewhat close to the problem at hand, at least in my opinion, what is about the intuition.

Recall that Lagrange's Four Squares Theorem asserts that every non-negative number is the sum of four squares of integers. Or in other words, for $L= X_1^2 + X_2^2 + X_3^2 + X_4^2$ one has $L(\mathbb{Z}^4)=\mathbb{N}$ (where we choose the convention that $\mathbb{N}$ includes $0$).

Now, this is on the one hand an interesting result in its own right, and on the other hand the fact that one can characterize the non-negative integers among all integers in such a way/by such a formula is sometimes also used. In fact, it is used in considerations close to the one at hand see its mention in the context of Hilbert's tenth problem.

So this is fine and interesting. What however does not seem like a very feasible idea is to try to understand the non-negative integers better by analysing the polynomial $X_1^2 + X_2^2 + X_3^2 + X_4^2$.

And, to some extent the situation for the primes and this polynomial seems comparable. Yes, one has this characterization of primes, this is very interesting but not so interesting to understand the primes (in the sense, say, some analytic number theorist would like to understand them, frequency, gaps, etc.): the description is not very convenient to work with, the primes do not come out in a systematic way, there are various polynomials having the same property (so why this and not another), the fact that such a polynomial exists is nothing very specific to the primes but also true for all recursively enumerable sets, and so on.

As remarked in comments it is always hard to say that something is definitely not useful in particular if the use refers to a vague notion as understanding the primes better.

Added to address additional/clarifies questions.

a. As mentioned below there are various 'formulas' that will yield arbitrarily large primes, however, they are not efficient in a certain way. Personally, I doubt one can get one (or at least a 'better' one) from considering this polynomial. Now, perhaps, I am wrong. I would be curious to learn if this were the case.

b. For Green-Tao and related results the following as some 'warning': The sixty-first Putnam competion (2000) had the following question (paraphrasing): the values the polynomial $Q=X^2+Y^2$ takes on $\mathbb{Z}^2$, contains inifinitely many triples of consecutive integers. This is not hard to show. Yet, then in the book "The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary" by Kedlaya, Poonen, Vakil there is mentioned the related question (p.279), (then) stated as open problem (again prarphrasing):

For $Q=X^2+Y^2$ the set $Q(\mathbb{Z}^2)$ contains arbitrarily long arithmetic progressions.

So if one cannot do this (directly) for this polynomial I am rather doubtful one can expect much for the one given here. One should note that, now, this problem is solved, but only (at least this is the 'official' solution on Kedlaya's webpage) via the result of Green and Tao [the set contains all primes congruent $1$ mod $4$, this is a set of positive relative density in the primes and thus it contains arbitrarily long arithmetic progressions].

As remarked in comments it is always hard to say that something is definitely not useful in particular if the use refers to a vague notion as understanding the primes better.