Pertaining to the answer within link.
Why is it the case, that for Lorentz invariant Lagrangian $\mathcal{L}$, after Wick rotation, the $O(4)$ invariance is established, thus manifesting itself as having Euclidean metric? Is that a consequence of requiring the four vector fields to transform as $A_0^E = iA_0$ and $A_j^E=E_j$ or a result of it? So which premise comes first?
As long as the Minkowski action is constructed from Lorentz-covariant tensors, then under Wick rotation [where the contravariant and covariant $0$-components of the tensors are Wick-rotated in opposite ways], the corresponding Euclidean action becomes constructed from the corresponding $O(4)$-covariant tensors, cf. e.g. this Phys.SE post.
Note in particular that the Minkowski metric tensor [with the signature convention $(-,+,+,+)$] is Wick rotated to the Euclidean metric tensor.