System F isn't consistent as a logic (pretty much precisely because it has general recursion). In languages with general recursion, you can do things like

(Haskell)

    anyType :: a
    anyType = anyType

or

(Rust)

    fn any_type<T>() -> T {
        any_type()
    }

System F doesn't have general recursion.

Extensions with a letrec-like construct are common, and are sometimes inaccurately called 'System F', but those languages do not have the properties of System F.

You're right. I must have been thinking of one of those extensions you're talking about (F# maybe?). I should have remembered that System F is part of the lambda cube, so it's at least as consistent as CoC