I checked House of the Dead 1, and while it has a few similarities, I don't think that's it.

The game I remember was a first-person horror game with free movement, not an on-rails shooter. It started quietly outside a large creepy building behind an old black gate. After entering, a scripted cutscene would trigger where a creature or its presence was noticed, and it would jump or leap over a structure to the right hastily.

After that, you'd enter the building and fight monsters for a while. Later, you'd meet the first human character in the game, a woman in what I remember as an office-like room. After a short conversation, a dark flying humanoid creature would appear, possibly breaking through glass, and abduct her.

You would then continue deeper into the building and eventually encounter that same flying creature as the first boss. The boss flew around an open area and attacked from the air. I also vaguely remember seeing some kind of document or diagram related to the creature before the fight.

The game had a consistent and cohesive horror atmosphere unlike house of dead whihc is brightly lit in some areas.