[PC][2000-2010]Hobo fight simulator
It was on PC but I think also on couple of other platforms. You were a detective who pursued serial killer killing other serial killers. Also there were prominently displayed birds in every episode. City had an extreme heat wave and that was "driving people crazy".
You were searching for that serial killer killer in all kind of desolate places and almost whole game you had only melee weapons and all kind of hobos were attacking you, usually also with melee weapons.
Most of the game was dark and you would hear hobos grunting and attack them that way.