It's the pve
I believe the people playing destiny spend more time in PVE situations than pvp (unless it's Iron Banner or Trial of Osiris time)
Halo needs fire fight
Saying it's the PVE is overly simplistic.
The key features that most of the successful games - not just shooters, but games, period - have is persistence and progression.
For Destiny, the persistent and progression is provided via the loot grind. Everything you do in the game, in every mode, goes toward RNG rolls, or towards building rep with factions (which lead to more RNG rolls), or getting specific pieces of gear that they can carry from that point on both as a sign of cred to other players, and to provide actual benefit in gameplay.
For COD, it's the level grind. New weapons, add-ons, perks, prestiges etc.
Both feed into one, single, persistent online identity. Every match, or game, contributes to it in some way.
The only persistent progression in Halo 5 exists in your stat profile and your REQ collection. The latter only has in-game consequence in Warzone. Firefight could help by giving players another mode to earn and use their REQ cards, (and frankly, so would some kind of scoring mode in campaign that allowed for REQ card use).
But that's not really as much of a draw, because you don't really 'own' anything. In Destiny, you get a cool drop, you don't only get so many uses until RNG blesses you again. In Halo 5, you use up all your Whiplash cards, you're SOL. Outside of the loadout weapons, there's no real sense of significant unlocking. That's part of the reason why the simple REQ 6 DMR is more coveted than the most powerful REQ 8 power-weapons (along with the DMR just being great for large maps, obviously).
Truthfully, the piece of persistence and progression that most players care about - particularly those that spend more time in Arena, is their stat profile. Their rankings, W/L, K/D, etc. That's where they see all their time and investment actually making a persistent impact. And that's what Bungie wanted, which is why the player's stat page is as easy to view in Halo 5 as the player's gear loadout is to view in Destiny.
They wanted Halo to be purely competitive. Which is fine. But if you're going to push that as your identity, then you better damn sure have the complimentary systems in place to ensure new players or lapsed players can get into the progression at any stage. Hook them and keep them. And frankly, the matchmaking and ranking systems are so disastrously bad, that they're failing to do so. COD and Destiny have the benefit that their progression and persistence draws don't require tight matchmaking. You get your ass kicked in COD or Destiny, you're still gaining XP toward that new unlock, or a chance at a nice loot drop.
In Halo 5, the persistence is wrapped up in the competitive pursuit, and they've absolutely dropped the ball at ensuring every match is competitive and fun for the player. The carrot-on-the-stick for Halo is watching that K/D and W/L increase. When you're constantly getting your ass kicked in one-sided mismatches, then the carrot gets further and further away, and you say 'fuck it'.
PVE will help. But it's not the fix many people are claiming it would be.
There is no way short of completely shifting philosophy and focus away from the purely competitive toward some other form of persistent progression, or completely overhauling the poor matchmaking and ranking systems that Halo 5 can ever hope to move up into the top 10 most played games on Xbox Live.