The Elliott-Halberstam conjecture is not known to follow from GRH.
Even the weak version of EH (which is with $Q=x^{1/2+\epsilon}$ for any fixed $\epsilon>0$)
does not follow from GRH. On the other hand, it is known that the Elliott-Halberstam conjecture almost implies the twin primes conjecture, i.e., it implies that there are infinitely many pairs of primes at distance $≤ 16$ (now $\le 12$, see Sylvain's comment). Furthermore, the Bombieri-Vinogradov theorem is indeed an amazingly strong unconditional replacement for
the GRH bound (and has as natural strengthening the EH conjecture).