^They have better PR because anime/manga always have them and anime/manga are more likely to have ridiculous feats of awesome and win than equivalent western productions. About the most badass thing anyone will ever do in western animations/comic books is cut someone in half with one swing while anime
regularly has guys cut buildings in half with single handed swings. For example, in Conan he cuts people in half with a bigass claymore and that's lauded as an accomplishment (it is in real life...). In Bleach, Zaraki Kenpachi quite casually
STABS a building in half using one hand and in the various Final Fantasy 7 spinoffs, Sephiroth cuts through skyscrapers singlehandedly with his 8 foot katana like they're not even there and repeatedly slices through a kilometer long cannon while floating 100 feet above it
with his sword lines.
I'm not quite sure why we're even debating any kind of sword's effectiveness against armor given that the only thing in this game with anything even remotely close to armor is the fleshpound... which gets decapped by a knife

Given that all zeds are unarmored, I would easily take any kind of sword over a fire ax and definitely a chainsaw which is damn heavy, impractical, and fairly dangerous to the user.