I have a mixed history with a lot of my early consoles. They were mostly gifts, and gifts I didn't ask for, so I didn't really get them in order to play any specific games. So beginning with the consoles I did get specifically just to play games:
PC: I smooth talked my way into getting my own PC in middle school. The parents knew I'd need to use the home computer more and more to do homework, and they were probably tired of me monopolizing the family computer. But the big secret reason I wanted my own was to play Counter-Strike on hardware that wasn't dogshit. When you learn how to play a skill based game at ~18fps, and suddenly get 120fps overnight, it became elucidating how important system specs were to my success. I went from struggling to get a 1:1 kd ratio, to becoming a pup-stomp lord who could get 10:1 in my sleep. TFC and simulation games like Sim City were also big factors, but CS was the big one.
Nintendo 64: This one was 100% Goldeneye. A friend invited me to a youth sleep-over at his church one evening, and me and him basically ignored everything and everyone and played Goldeneye all night long. It was mega addicting. I had previously sworn off console gaming before that, but playing 4 player Goldeneye tournaments brought me back into the fold. After that night, I moved hell and earth to get one at home. The N64 was a pretty barren machine, but Goldeneye was enough, and Zelda OoT was perennially on the horizon to convince me to keep the faith.
PlayStation: I was a big Nintendo fan back during this era, and had a lot of strong opinions about Sony's hardware decisions and ecosystem like a lot of you undoubtedly did as well. An uncle gave me a PS1 for Xmas one year though (I think it was x-mas of '98) and I was revolted by it. But it was free, so I kept it. I rented Final Fantasy VII because a few of my friends were big into console RPGs during the SNES Era, and kept singing the praises of the FF series, so I tried it out. Had a blast. But that was just one game, and I wasn't a convert yet. After doing some rudamentary research for more games like FF7, I came across Xenogears. And that game became my system seller. Because it's what convinced me to stick around in the Sony ecosystem and that these kinds of games just weren't possible on Nintendo hardware at the time.
Dreamcast: I don't know if there was any one game that sold me on Dreamcast. Especially when I had a pretty unfavorable opinion of Sega consoles and properties prior. But what ended up selling me on the console was the Dreamcast launch blowout issue of EGM back in '99. They made the console and its extensive launch library look amazing. I read that issue cover to cover multiple times and saved up my pennies doing chores and shit in order to buy one for myself by x-mas of '99. Not all of the games delivered, but Sonic Adventure, Soul Calibur, NFL 2K, Power Stone, etc were all amaze.
PlayStation 2: I preordered one of these for no specific reason, other than brand loyalty. I still didn't like Sony's design philosophy, but all the good games were obviously going to be on PlayStation, so I had to get one too. PS2 was a barren wasteland at launch, but the best launch game I remember getting was Dynasty Warriors 2 and Tekken Tag Tournament. My PS2 was basically just a machine to play PS1 games with marginally better load times for a good year, until Final Fantasy X and Metal Gear Solid 2 came out. Those two games were my white whales and convinced me to get the machine to begin with. But what was probably the most important game for me to stick with PlayStation was Zone of the Enders and The Bouncer - man those two games were incredible back in 2001, and both served as important previews for what to expect once FFX and MGS2 eventually came out.
Gamecube: I bought this machine out of brand loyalty, but was pretty disappointed by almost all the early games from it. "Is this really all this machine has?" while playing Super Monkey Ball. Mario Sunshine looked great, but I found the gameplay too obtuse to get into. It wasn't really until Soul Calibur II that I got a game that really wowed me. (Not even a system exclusive!) I held onto the machine for the promise of Wind Waker, and that's still to this day, probably the best game on the machine and the only reason to get one. That and Mario Kart: Double Dash - the best version of Mario Kart ever.
X-Box: I've never liked Microsoft machines, I find their business practices and console designs to be abhorrent. I went out of my way to avoid the X-Box. It wasn't until graduation day of High School that I finally played one, when a friend brought his over to another friend's party to play some Halo: Combat Evolved. And yeah, Halo was good shit. In the beginning of college, my friends were obsessed with playing Halo over LAN, so we all eventually ended up either getting our own X-Boxes, or living with someone who had one. It was that same feeling I had while playing Goldeneye back in the day, but with way better mechanics and multiplayer. All of my friends IRL are still trapped in the Microsoft ecosystem because of the legacy of Halo.
Xbox 360: Pretty much only picked one of these up to play Halo 3 with the friends. The first Xbox was an exclusively
Halo/Halo 2 machine. This one became a Halo 3 machine. My friends eventually moved onto Call of Duty/Modern Warfare, but I tapped out of that generation and sold off my 360 before it could RRoD because I was getting sick of playing the same Halo game over and over, and XBL/FPS games tend to bring out toxicity in me and my friends and I was getting tired of it. Bought a second 360 some years later to play Halo 4, regretted it after about a year and got rid of that Xbox too.
PlayStation 3: This was a brand loyalty purchase. I don't regret it in the slightest, because I still think PS3 had the best exclusives of that generation. But I bought this machine solely on the promise of Metal Gear Solid 4. Which ended up delivering on the hype, but that was a tough wait to sit through. Early games that bridged the gap in the meantime like Uncharted definitely helped make the wait bearable.
Wii: My friends IRL were completely captivated by the waggle-gimmick, but I saw past it pretty fast. I made it a point to get Twilight Princess for the Gamecube, not the Wii, because I wanted to play the game how it was meant to be played. Wii Sports was a lot of fun for a few evenings, but I ended up getting bored of it pretty fast. What made the system for me, was the promise of Super Smash Bros Brawl - which my friends and I got a lot of mileage out of. That and Virtual Console early on.
Nintendo DS: Brain Age, Professor Layton, and Wario Ware all helped justify buying in on the DS early. But it wasn't until Ace Attorney until the DS found itself a killer app for me.
PlayStation Portable: Three words: Metal Gear Acid. If you can't tell, I like Metal Gear games.
Nintendo 3DS: I skipped purchasing a DSi or any of the other upgrades for the DS, so getting a 3DS was mostly about future-proofing myself, and getting a replacement for my ailing fat DS. There weren't really any killer titles for the 3DS I was excited about until Fire Emblem Awakening.
Wii U: Honestly, this was the biggest waste of a console I've owned. I only played a small handful of games on it. Most of its killer apps either didn't appeal to me (all the Mario games; Splatoon) or you could play elsewhere. I basically bought it for more Smash Bros. It wasn't until Breath of the Wild that I finally sunk my teeth into a WiiU game with any real vim and vigor.
PlayStation Vita: Vita was worth it alone for it playing PS1 and PSP games on that gorgeous OLED screen. So I guess that was my system seller initially. Uncharted: Golden Abyss and Assassin's Creed: Liberation both convinced me the system had legs with regards to putting out good graphics/gameplay in your pocket. But it wasn't until Zero Escape: Virtue's Last Reward that I found a killer app for it.
PlayStation 4: Strangely enough, Assassin's Creed 4: Black Flag was my system seller for the PS4. I knew I'd want one eventually, but I went in on it early because I wanted to play the new Asscreed in the most optimal way (short of buying a new PC) and it delivered.
Xbox One: Bought this one to play Halo 5 and Modern Warfare/Call of Duty with the friends. Ended up regretting it again after a year or so and selling it.
Nintendo Switch: You can tell I've become old and boring when every new machine is bought to re-buy the same game over and over again. Smash Bros was yet again a system seller for me. But unlike the previous Nintendo consoles, this one has mostly delivered on not being a waste of money, since it's become home to all of the RPGs and quirky Japanese games that the DS and 3DS had captured in previous gaming cycles.
PlayStation 5: The immediate draw of the PS5 is this copy of Assassin's Creed: Valhalla I've got here. But I generally have a pretty big backlog of PS4 games I want to get around to/finish, and might as well play them on a better system where the framerate, resolution, and loading times can be pumped up.