"It appears to the naïve (me) that there is no nontrivial restriction on the primes for which we know there remain an infinite number in the restricted sequence.
Q1. Is this superficial perception in fact true?"

No, it is definitely not true. Here are a few examples:

(1) Let $a>0, b$ be two relatively prime integers. Is there infinitely many prime of the form $an+b$? Yes.

(2) Let $P(X)$ be a monic polynomial of degree $n$ with coefficients in $n$. Is there infinitely many prime $p$ such that $P(x)$ has $n$ distinct roots mod $p$? Yes.

(3) Let $X$ be a projective smooth variety over $\mathbb{Q}$, $\chi$ the Euler-Poincaré characteristic of the manifold $X(\mathbb{C})$, $n$ an integer. Is there infinitely many prime $p$ such that the 
the number of points of $X(\mathbb{F}_p)$ is $\chi$ modulo $n$? Yes. Same question with $X(\mathbb{C})$ replaced by $X(\mathbb{R})$? Yes.

(4) Is there infinitely many primes $p$ that can be written $a^2+b^2$? Yes. $a^2+8b^2$ with $b$ odd? Yes... 

(5) Let $a,b$ be two integers, $x$ and $y$  two reals between $-1$ and $1$, and $x$ smaller than $y$,  $a_p$ the number of solutions of $y^2=x^3+ax+b$ modulo $p$, minus $p$. Is there infinitely many primes $p$ such that $a_p=0$? yes.
Is there infinitely many primes $p$ such that $a_p \neq 0$? yes. Is there infinitely many
primes $p$ such that $x< a_p/2 \sqrt{p} < y$? Yes. 

One could multiply those examples. They all belong to algebraic number theory, 
and a line of thought that has begun with Dirichlet's theorem (example 1), and has developed into the modern theory of algebraic number field, Galois representation, automorphic forms and the Langlands program. Perhaps the most salient result is Cebotarev's density theorem, of which (1) is a very special case, (2) is a consequence, (3) also a consequence 
in combination with Grothendieck's étale cohomology, (4) also a consequence. Only (5)
lies really beyond this result, due respectively to Noam Elkies, Jean-Pierre Serre,
and the long list of people responsible for the proof of Sato-Tate. 

Admittedly, there are many natural and interesting 
sequences of integer in which we can reasonably conjecture that there is infinitely many primes, and to which this line of thought is not supposed to apply (Mersenne's primes, 
to name one).