Genre purism is a dead end. The arguably unsurpassed Mario 64 is a "platformer" that isn't about "platforming" at all. It's about completing objectives in a semi-sandbox. (GoldenEye was directly influenced by Mario 64, and this trickled down into Perfect Dark.)
Perfect Dark is an objective-driven FPS game. You are presented with objectives, and you complete them. This involves a lot of discovery and intuition. Sometimes problems have multiple solutions. Extremely similar gameplay loop to the Hitman games. The Hitman games are third person shooters that are not specifically about shooting. Nor are they about stealth in the traditional sense.
For its own part, Half-Life 2 is a story-driven FPS game which frames itself around a journey through a variety of locales. You shoot people because they are preventing you from reaching your destination. Shooting is not really your primary means of interaction with the world, but rather traversal, puzzle solving, and even driving.
FPS games where all you do is shoot are a genre of their own, to some extent, and that style of game is arguably better suited to multiplayer than singleplayer.
This is not a new disagreement. Doom was supposed to be story-driven with multiple male and female protagonists and a storyline that gave the player reasons to go from A to B. (Honestly, Tom Hall's design document paints a picture more akin to Prey than the Doom we got. Complete with interactive computer terminals and male/female lead choices.) Tom Hall, the original lead designer of Doom, strongly disagreed with the direction John Romero took the project, where the player spent most of the game engaged in shooting with no point. No character motivations. Not to say Doom is by any means a bad game, but the "no plot, shoot lots of stuff" design template is one of those things that suits a particular style of game. Half-Life 2 would completely collapse without its narrative structure. It simply wouldn't work as a game. Same goes for almost every modern FPS games.