You ask two different questions, I believe.

1. When Grothendieck invented étale cohomology, cohomology of coherent sheaves was already well known, thanks to Oka/Cartan's theorems A and B, Serre's FAC, Gaga, and the Serre-Grothendieck duality theorems). It was also well known (Weil, already) that the invention of some cohomology theory closer to singular cohomology was one path towards the Weil conjecture. In some sense, cohomology of coherent sheaves was an insufficient prototype.

2. Coherent sheaves naturally furnish examples of étale sheaves and Grothendieck's descent theorems imply that nothing is gained by viewing them in the étale framework. This implies in particular that étale line bundles or étale vector bundles are the same objects than line bundles or vector bundles. This gives at least some étale sheaves whose cohomology can be computed, such as $$ H^1_{\rm ét}(X,\mathbf G_m)=H^1_{\rm Zar}(X,\mathbf G_m)=\mathop{\rm Pic}(X).$$

For $n$ invertible on $X$, the Kummer exact sequence $$ 1\to (\mu_n)_X \to (\mathbf G_m)_X\xrightarrow{x\mapsto x^n}(\mathbf G_m)_X\to 1 $$ gives rise to a long exact sequence which is the starting path to the computation of étale cohomology groups with coefficients in $\mu_n$.

When $X$ is an $\mathbf F_p$-scheme, the Artin-Schreier exact sequence $$ 0 \to (\mathbf Z/p\mathbf Z)_X \to (\mathbf G_a)_X \xrightarrow{x\mapsto x^p-x} (\mathbf G_a)_X\to 0 $$ allows to compute the étale cohomology groups of $X$ with coefficients in $\mathbf Z/p\mathbf Z$. One sees in particular that they vanish beyond the dimension of $X$, hence are insufficient for the Weil conjectures.

You ask two different questions, I believe.

1. When Grothendieck invented étale cohomology, cohomology of coherent sheaves was already well known, thanks to Oka/Cartan's theorems A and B, Serre's FAC, Gaga, and the Serre-Grothendieck duality theorems). It was also well known (Weil, already) that the invention of some cohomology theory closer to singular cohomology was one path towards the Weil conjecture. In some sense, cohomology of coherent sheaves was an insufficient prototype.

2. Coherent sheaves naturally furnish examples of étale sheaves and Grothendieck's descent theorems imply that nothing is gained by viewing them in the étale framework. This implies in particular that étale line bundles or étale vector bundles are the same objects than line bundles or vector bundles. This gives at least some étale sheaves, such as $(\mathbf G_m)_X$ or $\mathrm{GL}(n)_X$, whose cohomology can be computed; for example: $$ H^1_{\rm ét}(X,\mathbf G_m)=H^1_{\rm Zar}(X,\mathbf G_m)=\mathop{\rm Pic}(X).$$

For $n$ invertible on $X$, the Kummer exact sequence $$ 1\to (\mu_n)_X \to (\mathbf G_m)_X\xrightarrow{x\mapsto x^n}(\mathbf G_m)_X\to 1 $$ gives rise to a long exact sequence which is the starting path to the computation of étale cohomology groups with coefficients in $\mu_n$.

When $X$ is an $\mathbf F_p$-scheme, the Artin-Schreier exact sequence $$ 0 \to (\mathbf Z/p\mathbf Z)_X \to (\mathbf G_a)_X \xrightarrow{x\mapsto x^p-x} (\mathbf G_a)_X\to 0 $$ allows to compute the étale cohomology groups of $X$ with coefficients in $\mathbf Z/p\mathbf Z$. One sees in particular that they vanish beyond the dimension of $X$, hence are insufficient for the Weil conjectures.

You ask two different questions, I believe.

1. When Grothendieck invented étale cohomology, cohomology of coherent sheaves was already well known, thanks to Oka/Cartan's theorems A and B, Serre's FAC, Gaga, and the Serre-Grothendieck duality theorems). It was also well known (Weil, already) that the invention of some cohomology theory closer to singular cohomology was one path towards the Weil conjecture. In some sense, cohomology of coherent sheaves was an insufficient prototype.

2. Coherent sheaves naturally furnish examples of étale sheaves and Grothendieck's descent theorems imply that nothing is gained by viewing them in the étale framework. This implies in particular that étale line bundles or étale vector bundles are the same objects than line bundles or vector bundles. This gives at least some étale sheaves whose cohomology can be computed, such as $$ H^1_{\rm ét}(X,\mathbf G_m)=H^1_{\rm Zar}(X,\mathbf G_m)=\mathop{\rm Pic}(X).$$ This generalizes to Zariski sheaves represented by a smooth group scheme.

For $n$ invertible on $X$, the Kummer exact sequence $$ 1\to (\mu_n)_X \to (\mathbf G_m)_X\xrightarrow{x\mapsto x^n}(\mathbf G_m)_X\to 1 $$ gives rise to a long exact sequence which is the starting path to the computation of étale cohomology groups with coefficients in $\mu_n$.

When $X$ is an $\mathbf F_p$-scheme, the Artin-Schreier exact sequence $$ 0 \to (\mathbf Z/p\mathbf Z)_X \to (\mathbf G_a)_X \xrightarrow{x\mapsto x^p-x} (\mathbf G_a)_X\to 0 $$ allows to compute the étale cohomology groups of $X$ with coefficients in $\mathbf Z/p\mathbf Z$. One sees in particular that they vanish beyond the dimension of $X$, hence are insufficient for the Weil conjectures.

You ask two different questions, I believe.

1. When Grothendieck invented étale cohomology, cohomology of coherent sheaves was already well known, thanks to Oka/Cartan's theorems A and B, Serre's FAC, Gaga, and the Serre-Grothendieck duality theorems). It was also well known (Weil, already) that the invention of some cohomology theory closer to singular cohomology was one path towards the Weil conjecture. In some sense, cohomology of coherent sheaves was an insufficient prototype.

2. Coherent sheaves naturally furnish examples of étale sheaves and Grothendieck's descent theorems imply that nothing is gained by viewing them in the étale framework. This implies in particular that étale line bundles or étale vector bundles are the same objects than line bundles or vector bundles. This gives at least some étale sheaves whose cohomology can be computed, such as $$ H^1_{\rm ét}(X,\mathbf G_m)=H^1_{\rm Zar}(X,\mathbf G_m)=\mathop{\rm Pic}(X).$$

For $n$ invertible on $X$, the Kummer exact sequence $$ 1\to (\mu_n)_X \to (\mathbf G_m)_X\xrightarrow{x\mapsto x^n}(\mathbf G_m)_X\to 1 $$ gives rise to a long exact sequence which is the starting path to the computation of étale cohomology groups with coefficients in $\mu_n$.

When $X$ is an $\mathbf F_p$-scheme, the Artin-Schreier exact sequence $$ 0 \to (\mathbf Z/p\mathbf Z)_X \to (\mathbf G_a)_X \xrightarrow{x\mapsto x^p-x} (\mathbf G_a)_X\to 0 $$ allows to compute the étale cohomology groups of $X$ with coefficients in $\mathbf Z/p\mathbf Z$. One sees in particular that they vanish beyond the dimension of $X$, hence are insufficient for the Weil conjectures.