This is very mysterious indeed. I remember I have also tried to find this result in Grothendieck's paper but it wasn't there, I think. Fortunately, you can find the proof in:

M. Zippin, Extension of bounded linear operators, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds., Elsevier, Amsterdam 2003, page 1725.

If you don't care about the norm of an extension of your compact operator, you can give a short proof of this theorem using the fact that the operation of taking the projective tensor product with $L_1(\mu)\cong C(K)^*$ respects quotients.

This is very mysterious indeed. I remember I have also tried to find this result in Grothendieck's paper but it wasn't there, I think. Fortunately, you can find the proof in:

M. Zippin, Extension of bounded linear operators, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds., Elsevier, Amsterdam 2003, page 1725.

If you don't care about the norm of an extension of your compact operator, you can give a short proof of this theorem using the fact that the operation of taking the projective tensor product with $L_1(\mu)\cong C(K)^*$ respects quotients.

Regarding your motivations, once you have your subspace $Y$, why can't you pass to a further subspace $Y_0\subset Y$ still having infinite codimension but additionally isomorphic to $c_0$ hence complemented? ($c_0$ is saturated by subspaces isomorphic to $c_0$.) Then you can extend $T|_{Y_0}$ to the whole space by 0.