There's three sides to this. A) They're just shoes, wear them; 2) Getting the leather uppers wet will them more susceptible to creasing, water marks, and require a shine after to look good; & iii) Dainite/Thames sole is a good compromise, since at least water won't seep upwards from the sole and soak the shoe.
I have a pair of EGs that I use as rain shoes, basically because I ordered them with the Thames sole. Occaisional polishing makes the leather upper plenty water resistant. Though they are black, and therefore much less prone to water spots.
In a hardcore downpour, I'd consider sneakers/rain boots however, no point in intentionally soaking through a pair of EGs.
P.S. They share rain duty with a pair of C&J brown suede shoes with a Dainite sole. Suede is great in the rain.