Halo didn't reinvent anything. It recycled a great deal and somehow made it very marketable. Its success prompted others to put out more FPS titles and particularly so for consoles. But it didn't do anything that other FPS titles hadn't already done much better, like Perfect Dark. And Perfect Dark only came to exist because it's the spiritual successor to GoldenEye 007. Halo didn't influence the genre so much as instill hope that copycats could sell well. For the most part in this way I think Halo spawned more weak FPS titles than anything else.
I feel like anyone voting Halo didn't pick up GoldenEye 007 when it came out. If anything, you're voting for the marketing and hype behind Halo more than the game itself.
Perfect Dark is one of my favorite games of all time and it is highly refined.
Then Perfect Dark Zero let me down because I was expecting quite a lot from the gameplay, but it was still a fun game.
Half-Life was huge, but not for its shooting mechanics or anything. It was good for the genre though.
Doom of course was a huge impact, but it didn't have the most impressive ingenuity or the greatest gameplay. And it didn't bring the genre to the console. Rare did all of that, in my opinion. There's a reason for the success of Perfect Dark remade for XBLA 10 years later.
I could be biased because I was completely unimpressed with Halo and then with Halo 2. I grudgingly beat Halo 2 on Legendary to find that yes, that WAS the real ending to the game.