This sounds like Darkest Dungeon.
I saw this video game years ago (less than 10 but it may have been older than that)
Released in 2016, so the timeline fits.
The animation was in the style of paper cutouts... You could never see the eyes of the protagonists as they were always covered by a shadow.
The characters and backgrounds look like 2D drawings. There is fluid movement when characters are walking, however the camera focuses on characters in static action poses during attacks and other events.
It is a dark and moody style, and if you look up images of the game you will notice all of the playable human characters' eyes are shaded or otherwise covered.
The game looked like it took place in the European medieval ages and the theme seemed to be one of cynical hopelessness and had a religious and magical theme.
The game is called 'Darkest Dungeon', and they deliver on both of those words. It's more of a 19th century European setting (flintlock pistols, highwaymen, Lovecraftian horrors, etc.), however the first area you explore feels like a classic medieval dungeon: stone walls, torches, and the occasional iron maiden.
The general theme is very dire, as it is set in and around a fallen mansion infested with horrid monsters, evil magic, and madness. The characters are rarely filled with hope, and your party members are expected to regularly die and be replaced with the next round of adventurers.
An enemy human (or mummy?) who was sided with the mutant animals yelled vaguely English-sounding gibberish at the protagonists (I remember one line started with an accusatory "YOU!" while pointing aggressively).
This is what made me sure it was Darkest Dungeon, because you perfectly describe an early enemy: the madman. He is in a tattered straitjacket that looks like mummy bandanges, and during his 'accusation' attack he screams and babbles incoherently, pointing at his target:
