You are corrent that Zellner's g-prior depends on the design matrix in the regression.  The prior is often used in "empirical Bayesian" analysis where there is no objection to this, but it is not a "pure" Bayesian procedure.  One counter-argument to this is that the design matrix is a conditioning object in regression, and so if we treat this object as a fixed value, then the dependence of the g-prior is no more than for other "objective" Bayesian procedures.  Nevertheless, I agree with your view that the prior is not compatible with the "pure" Bayesian approach, which would require the prior to be formulated without regard to the observed data.

As to the theoretical justification, a big part of it is just the mathematical convenience of using a conjugate prior, and the resulting simplicity of the posterior distribution.  The g-prior leads to a simple and tractable updating mechanism where the posterior mean is a weighted average of the prior mean and the OLS estimator, with a corresponding diminution of the variance.  There is some useful discussion in [Marayuma and George (2011)](https://projecteuclid.org/download/pdfview_1/euclid.aos/1324563354) with respect to the reasonableness of this prior, though this discussion is with respect to other criteria, not its independence from the data.