I won't use a synthetic, I don't even own one and have no plans to buy one. None of them that I've seen are anywhere close to the same type of quality as a mid-level leather saddle and I've rarely seen one that fits the horse wearing it.
Of course, though, I couldn't use a synthetic even if I did want to because I do a fair amount of roping and they aren't sturdy enough to stand up to that.
Anyway, as far as western saddles go, it really doesn't hurt them that much to get rained on. If you took the time to oil it properly when it was new, then most of the rain will just bead right off. You just have to take a few minutes and condition it where needed after it's dry. We've got western saddles in our tack room that are older than I am (29) and are still in great condition after spending decades riding countless horses in all types of weather.
I'd like to see someone claim the same of any synthetic.