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I believe that chapter $5$ of Galois Theories by Borceux and Janelidze contains what you're looking for -- they develop a general 'categorical Galois theorem' invoking toposes, then in chapter $6$ recover the special case of Grothendieck's theory for schemes using this categorical Galois theorem.

Chapter 7 gets even more general, presenting a 'non-Galoisian Galois theorem' that holds for any descent morphism in a category -- a morphism $f$ is a descent morphism iff the 'pullback along $f$' functor is monadic.

This paper by Christopher Townsend might be of interest to your specific question; he re-proves Joyal and Tierney’s result on the representation of Grothendieck toposes as localic groupoids as a consequence of a simplified case of the aforementioned categorical Galois theorem, then proceeds to actually prove the whole theorem using this trivial case as a key ingredient.

I believe that chapter $5$ of Galois Theories by Borceux and Janelidze contains what you're looking for -- they develop a general 'categorical Galois theorem' invoking toposes.

Chapter 7 gets even more general, presenting a 'non-Galoisian Galois theorem' that holds for any descent morphism in a category -- a morphism $f$ is a descent morphism iff the 'pullback along $f$' functor is monadic.

There is also the classic paper by Joyal and Tierney, An Extension of the Galois theory of Grothendieck, where they prove that each Grothendieck topos is equivalent to the category of equivariant sheaves on a groupoid internal to the category of locales.

This paper by Christopher Townsend might be of interest to your specific question; he re-proves Joyal and Tierney’s result on the representation of Grothendieck toposes as localic groupoids using a simplified case of the aforementioned categorical Galois theorem, then proceeds to actually prove the whole theorem using this trivial case as a key ingredient.

I believe that chapter $5$ of Galois Theories by Borceux and Janelidze contains what you're looking for -- they develop a general 'categorical Galois theorem' invoking toposes, then in chapter $6$ recover the special case of Grothendieck's theory for schemes using this categorical Galois theorem.

Chapter 7 gets even more general, presenting a 'non-Galoisian Galois theorem' that holds for any descent morphism in a category -- a morphism $f$ is a descent morphism iff the 'pullback along $f$' functor is monadic.

I believe that chapter $5$ of Galois Theories by Borceux and Janelidze contains what you're looking for -- they develop a general 'categorical Galois theorem' invoking toposes, then in chapter $6$ recover the special case of Grothendieck's theory for schemes using this categorical Galois theorem.

Chapter 7 gets even more general, presenting a 'non-Galoisian Galois theorem' that holds for any descent morphism in a category -- a morphism $f$ is a descent morphism iff the 'pullback along $f$' functor is monadic.

This paper by Christopher Townsend might be of interest to your specific question; he re-proves Joyal and Tierney’s result on the representation of Grothendieck toposes as localic groupoids as a consequence of a simplified case of the aforementioned categorical Galois theorem, then proceeds to actually prove the whole theorem using this trivial case as a key ingredient.

I believe that chapter $5$ of Galois Theories by Borceux and Janelidze contains what you're looking for -- they develop a general 'categorical Galois theorem' invoking toposes, then in chapter $6$ recover the special case of Grothendieck's theory for covering spaces using this categorical Galois theorem.

Chapter 7 gets even more general, presenting a 'non-Galoisian Galois theorem' that holds for any descent morphism in a category -- a morphism $f$ is a descent morphism iff the 'pullback along $f$' functor is monadic.

I believe that chapter $5$ of Galois Theories by Borceux and Janelidze contains what you're looking for -- they develop a general 'categorical Galois theorem' invoking toposes, then in chapter $6$ recover the special case of Grothendieck's theory for schemes using this categorical Galois theorem.

Chapter 7 gets even more general, presenting a 'non-Galoisian Galois theorem' that holds for any descent morphism in a category -- a morphism $f$ is a descent morphism iff the 'pullback along $f$' functor is monadic.