I suspect the support team think you still have an ADSL service, its the order people you need to be talking to. You haven't got a fault as such, just the replacement hub hasn't arrived.
The Argos one works fine, and even if you do eventually get a home hub, its useful to have a spare in case of storm damage.
If you want to avoid a lot of problems now and in the future get yourself a third party router. It will be superior to any ISP supplied rubbish.
You don't need an engineer out to tell you the HH4 won't work without an Openreach modem (which they no longer supply).
Save yourself the hassle.
As I said earlier, make and model is entirely up to you and your budget, you won't regret it though.