Well for what it's worth I think the people who will become famous today outside of the small universes of the art world or photography world are:

a) a few lucky people for whom a particular project takes off hugely in popularity
b) those who are photographing famous people, and then become famous by association

So outside of art and photography worlds no portrait photographer will become famous unless they take pictures of famous people or they fall into category a, which would be very rare.

Inside the "non photography-specific art world" it doesn't seem like the majority of photographers are concentrating on just portraiture, just landscapes, etc. Of course there are always exceptions to this, but most photographers seem to be working on mixed projects, and there is lots of fantastic portraiture happening in this context. But I wouldn't call those photographers "portrait photographers" per se. Nor are they particularly famous unless they also fall into category a.

Inside the photography world there are still lots of people who specialize in landscape, and there are lots of photographers who specialize in portraiture, but the audience for this work is even smaller than the already small "art world" so they will probably only ever be famous to people like us

Important Disclaimer: By spearating art world and photography world I am not at all suggesting that photography is not art, just trying to find a way to make a distinction between the kind of photography which is project based and currently in vogue in contemporary art circles, and the kind of photography which is popular in a more purely photographic context. ie Alec Soth is art world, Clyde Butcher is photography world. I love both of them for different reasons.

Here are some "art world" projects which mix portraiture and landscape and result in some great portraiture: