If only this were drawn differently, it’s got all the makings of a New Yorker style comic.
The expression really just ought to be don’t look a gift horse in the mouth until the givers leave. You wouldn’t want his dirty mouth infecting all your other ridiculous gift horses, now would you?
I can’t help but conflate this expression and the tale of the Trojan Horse. It’s sound advice: you really oughtn’t look into a gift Trojan Horse’s mouth lest you enjoy being riddled with arrows. Or martyrdom, I suppose. One ungracious Trojan would have saved the whole city, eh?