TITLE: Why so difficult to prove infinitely many restricted primes?
QUESTION [25 upvotes]: I wondered whether there were an infinite number of
palindromic primes written in binary (11, 101, 111, 10001, 11111, 1001001, 1101011, ...)
and quickly discovered that it is unknown
(OEIS A117697).
Indeed, even though almost all palindromes in any base are composite,
whether there are an infinite number of palindromic primes in any base is unknown
(Wolfram article).
Earlier
(in the MO question,
"Why are this operator’s primes the Sophie Germain primes?"),
I learned that
it is unknown if there are an infinite number of
Sophie Germain Primes.
In addition, is not known if there are an infinite number of
Mersenne primes,
Fibonacci primes (OEIS A005478),
Wilson primes,
Cullen primes,
not to mention prime twins, quadruplets, sextuplets, and
$k$-tuples.
No doubt this list of our ignorance could be extended.
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?
Q2. If so, is there any high-level reason why it is so difficult
to prove these statements?  Or is each difficult for its own idiosyncratic
reason?

I ask this out of curiosity, without expert knowledge of number theory.
Thanks for enlightening me!
Questions Answered. Thanks for the wonderfully rich and informative answers!
Essentially both questions have been answered: My superficial perception (Q1)
is not in fact accurate, as detailed in the examples provided by quid, Anthony Quas, and Joël,
augmented by comments by several.
A high-level reason (Q2) explaining the difficulty in the examples I listed was nicely encapsulated
by Frank Thorne, enriched by appended comments.  Thanks!

REPLY [23 votes]: "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. Are there infinitely many prime of the form $an+b$? Yes.
(2) Let $P(X)$ be a monic polynomial of degree $n$ with coefficients in $\mathbb{Z}$. Are  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. Are 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) Are 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 (such that $4a^3+27b^2 \neq 0$), $a_p$ the number of solutions of $y^2=x^3+ax+b$ modulo $p$, minus $p$. Are there infinitely many primes $p$ such that $a_p=0$? yes.
Are there infinitely many primes $p$ such that $a_p \neq 0$? Yes. Let $\alpha$ and $\beta$ be  two reals between $-1$ and $1$, and $\alpha$ smaller than $\beta$; are there infinitely many
primes $p$ such that $\alpha < a_p/2 \sqrt{p} < \beta$? 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 fields, Galois representations, 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 integers in which we can reasonably conjecture that there are infinitely many primes, and to which this line of thought is not supposed to apply (Mersenne's primes, 
to name one).