Both Scotland and England gave up their sovereign status when the Acts of Union were enacted, to form the United Kingdom of Great Britain.
Basically, Scotland, Wales, England, and Northern Ireland, are nations which comprise the United Kingdom of Great Britain and Northern Ireland, which is the sovereign state.
But to the topic, yes there's no apparently shorter way of referring to the UK as anything else but "Britain"
Other Kingdoms get away because they only have a "single name" (Netherlands, Spain, Sweden, etc)
Unless we come up with a name like the flag and call it Swaland or something like that
Sure, but in a sense where UK “countries” are a thing somewhat like US “states” (but somewhat less, in that neither is independent, but US states are notionally sovereign nevertheless.)
The UK is the entity which is a sovereign and independent entity.