People are interested in computing the zeros of $\zeta(s)$ and related functions not only as numerical support for RH. Going beyond RH, there are conjectures about the vertical distribution of the nontrivial zeros (after "unfolding" them to have average spacing 1, assuming they are on a vertical line to begin with).
Odlyzko found striking numerical support for such conjectures by making calculations with zeros very high up the critical line: hundreds of millions of zeros around the $10^{20}$-th zero. See the Katz--Sarnak article here and look at the picture on the second and fourth pages. These vertical distribution conjectures do not look convincing by working with low-lying zeros.
From Wikipedia's table on its RH page, the latest exhaustive numerical checks on RH (all zeros up to some height) go up to around the $10^{13}$-th zero. There are other conjectures that have been tested numerically far beyond $10^{13}$ data points, e.g., the $3x+1$ problem has been checked for all positive integers up to $80 \cdot 2^{60} \approx 10^{19}$, Goldbach's conjecture has been checked for the first $2 \cdot 10^{18}$ even numbers greater than $2$, and the number of twin prime pairs found so far is over $8\cdot 10^{14}$. With such examples in mind, I would not agree that the numerical testing of RH is out of line with how far people are willing to let their computers run to test other open problems.