My understanding is that all the merchants out there, even just the ones using Ingenico terminals, still have various payment services providers (PSPs) who would have to specifically choose to offer Bitcoin integration and push it to their client devices for what you describe to work. A Bitpay account is needed so that the terminal knows who to talk to (thus, why it needs a direct connection to the internet) and so that Bitpay knows who to pay (what business, what bank account).

Bitpay is also customizing the software on these devices themselves (loading their own app), thus why the suggested route is to order a terminal from them. It is unclear whether or not a merchant with an existing terminal can modify it themselves.