I love Legendary Pokemon. I find their stories fascinating (most of the time *looks at Heatran*), and I like the idea of these powerful, immortal and/or worshiped beings wanting to make friends with a trainer like any other Pokemon. I've never really had a problem with the games being too easy when using them (I find that levels are more important during the main story than the power of the Pokemon you're using), but I usually have a full team established by the time you can catch a Legendary, so I don't get to use them often. :/
The most notable exception was the Liberty Garden Victini in Pokemon Black - I used her on my main team throughout the game. Seriously, the Liberty Ticket event may have been the best handled event in the series (giving you the Pokemon early in the game, letting you catch it yourself, and coming with an interesting story), and the experience contributed to Victini being my (pre-gen 7) favorite Pokemon. Her name was Victoria. :3
I also swapped Reshiram into my team for the final battles, because destiny and whatnot. (Victoria was the one I swapped out, actually - a Fire type for a Fire type and a Legendary for a Legendary.)
I didn't know I was going to be getting a Latios in Omega Ruby, so I already had a full team at that time - I tried to rotate him and one of my party members, but it just didn't work out. I'd like to replay the game sometime and leave an empty slot. I named him Eos. (I know that's a feminine name, but it just seemed to fit.)
I played Pokemon Moon blind, but I just so happened to only have 5 Pokemon when it was time to catch Nebby - a coincidence that still boggles my mind. So, Moon-Deity-Nebby was also a member of my main team of 6, if only towards the end.
I did not use Nebbycrozma (Dawn Wings? What's that? 🙃) against the Elite 4 in Ultra Moon, despite it being my post-gen 7 favorite Pokemon; I already had to make room on my team for Naganadel (do Ultra Beasts count as Legendaries for the purposes of this question?), and anyway, it just wouldn't feel right to use experimental technology to merge two Pokemon into a new form of life and then immediately enter said form of life in a major tournament. However, I used them in the Rainbow Rocket episode (as well as Silvally and Stakataka). Forget the type disadvantage - using Light That Burns The Sky on Giovanni's Mewtwo was just the most epic feeling.
Finally, I used a glitch-obtained Mew, named Eve, in my playthrough of Blue on the Virtual Console (which was the first time I ever played gen 1). Honestly, having a Mew added a poignant layer to the story, because, as we all know, you can catch Mewtwo in the postgame. So, my character's ultimate goal in the story wasn't to become champion or defeat Team Rocket...but to reunite Eve with her child. D'aww.