Just realized this

But what if they do, and its just synced up with your blinking without you knowing it

They do, but the camera is positioned in front of their eyes.

In an FPS you're whatever character that you're playing as so by extension when you blink they're blinking too

...okay I think it's time for a quick fresh air break for everyone

You blink in the short first-person segments of MGSV

Alone in the dark

SCP Containment Breach.
Also, what @metalman said is what most game designers go with.

It's because you blink for them... Carrot