This is a somewhat subjective topic, but a lot of people believe that the answer is "no". There are various reasons why the RH for curves is much easier than the general case.

One is that for curves, one can replace $\ell$-adic cohomology with the Jacobian. In higher-dimensions the geometric objects underlying $\ell$-adic cohomology groups are motives.

Another is that in dimension 1, RH is equivalent to the estimate on the number of rational points $|\#X(\mathbb{F}_{q^r}) -q^r| = O(q^{r/2})$. This is exactly what Stepanov and Bombieri proved. But in dimension d>1 the main error term comes from the cohomology in dimension $2d-1$, and so a point-counting estimate does not give you any more information than the Lang-Weil estimate $|\#X(\mathbb{F}_{q^r}) -q^{dr}| = O(q^{(2d-1)r/2})$ - which is proved by reduction to curves.

There are some special cases where RH *is* equivalent to a point-counting estimate - for example, (smooth projective) hypersurfaces, where the only interesting cohomology is in the middle dimension. Katz asked a long time ago whether one could give an elementary proof in this case, and Bombieri also thought about it. (I recently found how to deduce the general RH from the case of hypersurfaces, so a different proof of this special case would certainly be interesting.)

Taking a step back, you can also ask for "simpler" proofs of the other parts of the Weil conjectures. The proof of the rationality of the zeta function for curves is very simple, just using Riemann-Roch. As far as I know, there are no simple proofs in dimension > 1, although a while back Fesenko mentioned in a paper that his adelic methods would give a non-cohomological proof of rationality for surfaces.