Big Bang and Inflation are't alternatives, they're part of the same theory. Inflation followed the Big Bang. In fact, recent studies of the CMB provide evidence for both the Big Bang and inflation.
As is typical when this happens, parts of the original theory remained. For example, the "big bang" event- the idea that everything started from an initial 'explosion' of space-time- remained. And the term is still used.
My understand is that this is also true with Darwin's Theory of Evolution- I am not a biologist however so I may be off-base here. The concept of evolution- specifically survival of the fittest and speciation- remain in the current theories. And people still speak of 'evolution.' But the accepted/currently debated theories differ in some fundamental aspects, notably the gradualness of the process.
So my point was that we will be speaking of 'black holes' for a long time even if the current 'Black Holes' as we think of them today are shown to not actually exist.