In Thomason's paper Algebraic K-theory and etale cohomology, Thomason develops an elaborate theory of Hypercohomology spectra, $\mathbb{H}(X,\mathcal{F})$ for presheafs of spectra $\mathcal{F}$ on the étale site of the scheme $X$.

As per my current understanding these Hypercohomology spectra come equipped with a 'descent' spectral sequence, which in the particular case of Algebraic $K$-Theory, considered as a presheaf of spectra on the étale site of $X$, starts from the étale cohomology (with Tate Twisted $\mathbf{Z}_l$ coefficients) of $X$ and converges to the localization of $K(X)$ with respect to mod $l$ complex $K$-theory. (We assume $l$ is invertible in $X$.)

My question is if there exists a more modern reformulation of this theory.

I am quite certain that since the paper was written in the $80'$s most of the techniques might have become standard and the results should follow from the homotopy theory of spectra valued presheafs on a site (in a more modern language).

I would be particularly interested in a reformulation of this work in the language of $\infty$-categories.