As kids, our parents typically didn't get us games at launch. We had to usually wait until birthdays or Christmas. That's just how it goes.
Our mom knew how much we wanted to play Super Smash Bros. Brawl and took us to a midnight release of it. We also got Mario Kart Wii on launch day with a massive trade-in deal from GameStop of some games we barely touched. We also got New Super Mario Bros. Wii on launch day, and Super Mario Galaxy 2.
Super Smash Bros. for Nintendo 3DS is one of the earliest ones I remember getting with money that wasn't leftover from a birthday or a special treat from my family.

I want to say it was Sonic 3 on the Genesis. But I am not 100% sure on that. It would have been a game my mom bought my sibling and I to play. As the youngest I always stuck playing the old Sonic games as Tails unless I was playing by myself. 

Pokemon yellow. When pokemon was actually good.

Pokémon Alpha Sapphire

I know Warcraft 3 for sure. I’m pretty sure all the other games I got were months after launch.

Never bought anything quite at launch.
Probably closest are Sonic 1 and Asterix on Master System. (Might be only games I've ever bought within a year from launch...)

If there were any before LoZ: Breath of the Wild I have forgotten. Odds are there were but the last I remember for sure is that one. Saw it in the direct and although I was iffy on the Switch after the WiiU I pre-ordered both the Switch and Breath. I wanted the best possible way to play that game.

The first game I ever bought was Mario sunshine with the gamecube. I remeber how many chores and odd jobs I had to do around the house and neibourhood to save up the money to buy it.

Gran Turismo 6 for the PS3 back in 2013.