The real question is how influential are they today? I think they might not be that influential when it comes to AAA titles, however, some outlets are extremely influential when it comes to indie gaming, since it's a small scene where "knowing the right person" can make a difference when it comes to coverage.
My problem with the press is that there is essentially 2 categories:
- marketing outlets AKA content farms such a IGN,Gamespot who add very little value to the content they produce as they seek quantity over quality.
- pseudo-intellectual outlets like Eurogamer or Polygon which talk about gaming through the lens of "intersectionality", which is their right, but after 10 hours of work getting yield at by my manager I'm not interested in articles that are more about identity politics than the games I play when I get home.
Personally, I take more pleasure in hearing Joseph Anderson babbling about hitboxes in Dark Souls 2 and how you get critical damages even though the NPC sword stroke was PI/2 radians off your character's position, for 3 hours straight on youtube than any of the former.