In Thomason's paper Algebraic K-theory and étale cohomology, (Ann. ENS 1985, Numdam link) 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 must 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.