Why do you run Nightfalls every week?
Why do you run Raids every week?
Why do you play in the Iron Banner?
Why do you do particular PvE/PvP tasks to complete quests?
Why will you want to complete the Lost Sectors and Adventures in Destiny 2?
Answer to all of the above is (at least initially) for the loot (just some examples of the activities but not all that are loot driven).
Yes, the gameplay and hopefully the exploration is a reason in and of itself to do these things but the loot drives the game.
And yes, there are (down) times when you basically have everything in the current iteration of Destiny where you'll be playing for simply the love of the gunplay/gameplay but you'll also be anticipating the day when new weapons/gear will be introduced and acquired in the next iteration.
Just because Destiny doesn't have the variation or amount of loot compared to other games (Borderlands, Diablo, The Division, etc) doesn't mean it's not a loot game.
Plus, Cayde-6 said it was.
1. I don't run it weekly unless I want the buff.
2. Again, I run Raids when I want to, for the content itself.
3. I play IB for the PvP.
4. I complete quests because that's literally the
content in the game.
5. I actually want to see the encounter designs and how the lore is comprised in world.
But really, the answer to all those questions is that people do those things because it's what you do to play the game. Run dungeons, Do patrols, do Quests, do dailies, PvE, PvP... all of it is the meat of the game. It's what you do in the game. If you're not doing those things, you're not playing the game. You're not even guaranteed loot from doing the majority of the content.
"Loot games" are not the only games with loot. They're not the first game with loot. RPG's started all of that. Hell the term Loot as it's used in gaming comes from TTRPGs. So Cayde-6 mentioning loot does not mean the game is a loot game. Just a community meme reference.
I mean, people can play the game for whatever reason suits them. Maybe it's the way the game feels as a shooter. Maybe it's the lore. Maybe it's the PvP. Maybe it's the loot. All of those are perfectly valid reasons to play. But saying a game that's identifiably a Lite MMO (Even Bungie's term Shared-World Shooter hints at MMO conventions) is a Loot Game is like saying Warcraft 3 is a PvE MOBA.