Let $X,Y$ be noetherian connected schemes over an algebraically closed field $k$ and let $\overline{x},\overline{y}$ be geometric points on them. There is a canonical homomorphism
$\pi_1(X \times Y,(\overline{x},\overline{y})) \to \pi_1(X,\overline{x}) \times \pi_1(Y,\overline{y})$
This is known to be an isomorphism when $X$ or $Y$ is proper and geometrically integer. This follows from a more general exact sequence in $\pi_1$ related to a proper flat morphism with geometrically integer fibers.
Question 1. Is there a more direct proof of this isomorphism? The corresponding isomorphism in topology is trivial, is it really that difficult in algebraic geometry?
Question 2. Is the assumption on $X$ or $Y$ really needed? If yes, can you give an easy counterexample for the general case?