Well, suppose pi is a cuspidal automorphic representation of GL(n)/Q. This has the structure of a tensor product, indexed by primes p, of representations pi_p of the groups GLn(Qp). The Satake isomorphism tells us that at almost all primes, each pip is determined by a conjugacy class A(p) in GLn(C). In this language, the Riemann hypothesis for the L-function associated to pi says that the partial sums of tr(A(p)) over p < X show "as much cancellation as possible," and are of size sqrt(X). But if n>1, we are dealing with very complicated objects, and the local components of these automorphic representations vary in some incomprehensible way...
You are right, there are certainly special cases. If we knew GRH for L-functions associated to Artin representations then the Cebotarev density theorem would follow with an optimal error term. Likewise, GRH for all the symmetric powers of a fixed elliptic curve E implies (and is in fact equivalent to; see Mazur's BAMS article for a reference) the Sato-Tate conjecture for E with an optimal error term. But in general, reformulations like this simply don't exist.
There are many interesting consequences of GRH for various families of automorphic L-functions. I recommend Iwaniec and Kowalski's book (Chapter 5), the paper "Low-lying zeros of families of L-functions" by Iwaniec-Luo-Sarnak, and Sarnak's article at http://www.claymath.org/millennium/Riemann_Hypothesis/Sarnak_RH.pdf
I'm not an expert in this area, but this may be a start.
Rather than $\prod_{p\lt n}$, you can use $\prod_{p\le \sqrt n}$.
$\log\log \sqrt n + \gamma \lt \log\log \sqrt n +\log 2 = \log\log n $
That gets you a little closer, since now you are off by $\log 2 - \gamma \approx 0.116$.
The heuristic probability that $n$ is prime is not
$$\prod_{p\lt n} (1-Pr(p|n))$$
It is the product of probabilities
$$\prod_{p\lt n} (1-Pr(p|n \text{ given no smaller prime divides } n))$$
For $p$ small, the term you get may be close to $(1-1/p)$, but I that's not the case for $p$ large.
For $\sqrt n \lt p$, the term corresponding to $p$ in the product is just $1$.
For $\sqrt[3]n \lt p \le \sqrt n$, if $p$ is the smallest prime dividing $n$, then $n/p$ must be prime, too. Perhaps that means that by strong induction, we should discount these terms by the probability $n/p$ is prime, about $1/\log \frac np$, so that those terms in the product are $(1-1/(p \log \frac np))$.
It looks like you get some sums/integrals if you try to extend this to more terms, and I don't know whether you can expect to get the desired accuracy at the end.
Best Answer
The Riemann hypothesis is true, if primes are random in certain ways.