It was always "always online" (though you could play matchmaking only activities solo pretty easily). But it didn't always have the traits of live service that destroyed it.

Destiny 1 had relatively little content, with excellent enemy design and level design, done in a way that the content had a lot of replayability purely for the sake of the mechanics. Playing a strike again (provided you weren't too overleveled and mowing through everything) never really felt the same because you and your party didn't do the exact same thing. This was supplemented with nightfalls (which were a live service like timed event, but didn't have to be) and other strikes with modifiers to expand the replayability more. Every once in a while there was an expansion, with a well designed new map, and you could still play all the old stuff. There were a handful of new type of enemies, playing on the old ones, that were still thought out and well structured.

Destiny 2 obviously threw out the D1 content. Fine. It's a new game. But, because of "not enough content" complaints, they constantly threw old stuff in the trash for rushed, badly designed new ones. They stopped focusing on well balanced strikes and focused on new game modes designed to serve the "season" model by forcing you to use specific weapon types to even do damage on certain enemies. They mostly threw the well balanced original enemies in the trash for poorly thought out new ones for the sake of "this is new" at a pace that made no sense.

New content being different is fine. Taking away old better content to force people into half assed content churn is not, and that's what the live service emphasis did. I'm not even sure the significantly worse content was an "innocent" consequence of the bad pace, either. The poor design significantly cut down replayability, meaning more players are clamoring more for new opportunities to throw money at new bad content.

They make a good shooter mechanically.

But all the great enemy design and map design that Destiny started with took a back seat to constant content churn with all the actually well designed stuff taken away and made impossible to play again.