I already explained to you how it can not be informed by the American frontier. Take a look at the D&D monster books. The overwhelming majority of them cannot exist in a world that has no uncivilized areas. There's no place for them to live where they wouldn't be instantly killed by the civilized races.
D&D by it's very nature requires on an academic level, uncivilized wilds where monsters dwell. That necessity automatically creates a frontier between the civilized areas and the wilds. At no point is the American frontier necessary to even know about for a frontier to be necessary for D&D.
You mentioned Warhammer not having a frontier. For that to be possible, there can be no uncivilized area anywhere that is not completely isolated, such as an island in the middle of the ocean. That means that unless you're going to set up nonsensical gameplay where dragons, beholders and Remorhazes just hang around next to towns and cities, you don't have that sort of monster in the game.
In my opinion it doesn't really matter. And in D&D it's all of those. At no point in the history of D&D has every frontier town been a lawless(or nearly so) place with saloons and spell fights happening over insults. Some may be like that, but others have very different feels to them.
As an American who grew up on cowboy movies and tales of the old west, I didn't get the western frontier vibe from the majority of D&D frontiers.
