Let $f(x):=ax+b$ for naturals $x,a,b$, such that $f(x)$ will take infinitely many primes as $x\to\infty$.
Is it the case that for any choice of $a,b$, there exists some $N$ such that for all $n>N$, $\pi(n)<\pi_{a,b}(n)$?
In other words, in the limit, do the naturals yield fewer primes than all admissible arithmetic progressions? This seems likely to be true heuristically, but it's non-obvious to me past that.
If you mean the first $n$ terms of $\{ax+b\}$, then yes.
For example, primes among the odd numbers are more common than primes among the integers.
They are all squeezed into series with b that are coprime to $a$. That increases the density by a factor $a/\phi(a)$.
On the other hand, numbers are larger by a factor $a$, so primes become rarer by a factor $(\ln n)/(\ln an)$.
This second factor approaches $1$, so eventually the $a/\phi(a)$ dominates, and each sub-sequence has more primes than the full sequence.