The game is Rainbow Six Vegas
And you can tell it’s 2006 because it has that distinct visual bloom from early 360/PS3 games where objects are extremely bright.
I’m surprised how competent the game is. It’s interesting playing a modern warfare shooter before COD4 was a thing.
It also has some of the same voice lines/actors as Alien Swarm, another tactical squad based shooter like Rainbow Six except it’s top-down. I assume Alien Swarm just used them or they’re stock lines.
The most disappointing part of this game is that it has no quick saves. You’re tied to where the devs place checkpoints, so if there’s a long stretch of difficult battles and you die, you have to do that all over again. At least it isn’t a stealth game.