Yes we tried that with New Street several decades ago
International is also not pleasant to use for similar reasons of low ceilings. I think I found changing there worse than changing at New Street, though neither was good and I'd far prefer other stations where possible.
A terminus is slightly different - there's typically less changing between trains, and you don't rarely wait on the platforms so the platforms don't have to be as pleasant. But still, I'm not sure you want to make it too terrible.
Hopefully the plans have opportunity for money making (beyond the concourses), as well as ways around said money making (Stratford International's mall is annoying if you just want to get your train, though you can bypass the crowds by going along the upper floor), but they cannot prioritise money making over providing a pleasant environment for train customers.
I think there was provision for a segregated international platform at one stage, but that probably went after abandonment of the HS2-HS1 link.
I've never seen any planned service pattern that had trains heading south from Curzon Street to anywhere but Euston. The two pencilled-in paths for Heathrow or HS1 served Birmingham at Interchange before heading further north.
It may come into use if HS4 Air becomes a thing.
But HS4Air services to Birmingham (ditto Manchester, Leeds, etc) require diverting existing Birmingham trains at the London end, therefore it won't require any extra trains terminating at Curzon Street (or Piccadilly), in the highly unlikely case that it the boondoggle is built.
This is where extra services would come from. NPR allowing High Speed Liverpool-Birmingham trains.