Cassell’s reports a 1930s usage of shitfaced meaning stupid or ignorant. No citations. Also a 1970s usage meaning complete or total, as in shitfaced drunk. So we may have a progression from “total idiot” to “total” to “totally drunk,” although I haven’t done enough research to be confident of this chain.
There is also the Briticism arseholed, also meaning drunk, which the OED dates to 1982. Partridge claims the phrase pissed as arseholes dates to the 1940s, but only provides evidence from 1961,
I think the “face in the toilet” explanation is reading a bit too much into it. Slang terms like this don’t need to make sense. The scatological reference is enough.