Objectively, this would be quite simple. Just pick the top four best selling games for each console.

NES:
Super Mario Bros., Super Mario 3, Legend of Zelda, Megaman 2

SNES:
Super Mario World, Street Fighter II, Super Mario Kart, DKC

N64:
Mario 64, GoldenEye 007, Zelda: OoT, Mario Kart 64

Sega Genesis:
Sonic 1, Sonic 2, Mortal Kombat 1, Madden NFL

Playstation 1:
Final Fantasy VII, Resident Evil 2, MGS, Gran Turismo

Sega Dreamcast:
Soul Calibur, Jet Set Radio, Powerstone, NBA 2K

PS2:
GTA3, GTA San Andreas, God of War, MGS2

GameCube:
Smash Melee, Mario Sunshine, Zelda: Wind Waker, Metroid Prime

XBox (OG):
Halo, Halo 2, Splinter Cell, Star Wars KOTOR

PS3:
MGS3, Uncharted 2, The Last of Us, GTAIV

XBox 360:
Call of Duty, Halo 3, GTAIV, Red Dead

Wii:
Wii Sports, Wii Fit, Super Mario Galaxy, Mario Kart Wii

Wii U:
Splatoon, NSMBU, Mario 3D World, Mario Kart 8

Switch:
Zelda: BOTW, Super Mario Odyssey, Mario Kart 8 Deluxe, Animal Crossing

PS4:
Spider-Man, GTAV, Call of Duty (various), God of War Ragnarok

Game Boy (OG):
Tetris, Super Mario Land, Zelda, Pokemon

GBA:
Advance Wars, Pokemon (various), Mario Kart Super Circuit, Fire Emblem

Nintendo DS:
Brain Training, Nintendogs, Mario 64 DS, The World Ends With You

Nintendo 3DS:
Kid Icarus Uprising, Mario 3D Land, Mario Kart 7, Pilotwings Resort

Those are the systems I care to list, the rest I just don't care enough about (sorry PSP and Vita and PS5 and XBox One) to have much of an opinion. 

That would mean a lot of COD for PS3 and 360 onward. 

Yes, it would lol. But when making a Mt. Rushmore, I feel like those games would actually be there if you polled everyone who gamed during those gens. People love what they love, even when it's junk like Fortnite.

PS1:
Final Fantasy 7, Tekken 3, Gran Turismo, Crash Bandicoot

PS2:
Grand Theft Auto: San Andreas, Shadow of the Colossus, God of War, Kingdom Hearts

PS3:
Uncharted 2, Grand Theft Auto 4, Grand Theft Auto 5, The Last of Us

PSP:
Tekken Dark Resurrection, LocoRoco, God of War: Ghost of Sparta, Patapon

PS4:
God of War 2018, Spider-Man, Uncharted 4, Bloodborne

PS Vita:
Persona 4 Golden, Tearaway, Gravity Rush, Little Big Planet

PS5:
Astro Bot, Demon's Souls Remake, God of War Ragnarok, Returnal

I'm going to try something different. Rather than highlighting the best or most popular games per console, I'm going to try selecting four games that illustrate the nature of a console's library.

NES

• Dragon Quest III - This is a bit of a Japan-focused pick, but it's hard to overstate the importance of the NES era Dragon Quest games on the tastes of the Japanese market. DQ3 in particular is the second-best selling third party game on NES, and was possibly the fastest-selling console game ever for a few years.
• Legend of Zelda - Along with being the 6th best-selling NES title, this game represents the Famicom Disk System and the emergence of games that supported saving, as well as games on the NES with a more open structure such as Metroid and early JRPG's.
• Super Mario Bros / Duck Hunt combo cartridge - This was the best-selling cartridge for the NES, and was bundled with a good portion of the console's library. It also gives me an excuse to include both the original Super Mario Bros to represent that series and Duck Hunt for games that used peripherals.
• Teenage Mutant Ninja Turtles - Regardless of whether the game is very good or not, this game (which is incidentally the best-selling non-Nintendo game on NES) represents how licensed games were a huge part of classic era console libraries. It also shows how even well-respected developers like Konami would often be behind the development of popular licenses. TMNT was technically published by "Ultra Games," a subsidiary of Konami exclusively made to get around Nintendo's regulations regarding limits to a publisher's releases per year.

SNES

• Chrono Trigger - Chrono Trigger is being used here primarily as a stand-in for the many RPG releases from companies like Square and Enix. Of the SNES's Top 25 known best-sellers worldwide, 8 are JRPG's, despite the genre still being relatively niche in much of the world. Chrono Trigger in particular is being used since its development included talents behind games from Square and Enix alike.
• F-Zero - F-Zero sold fairly well, #15 on the console overall, but is included here to represent how developers were using the SNES's new tech (such as Mode 7) when developing games. Mode 7 in particular was used in dozens of games, be it to create sprawling overworld maps in RPG's, create a pseudo-3D effect for racing games like Super Mario Kart, cinematics like in Super Metroid, etc. 
• Street Fighter II - The big third party game of the era, and one of the often forgotten reasons the SNES caught up to the Genesis worldwide. Street Fighter II's base game was the best-selling third party game on a Nintendo platform until 2010's Just Dance 2, and was the vanguard for the fighting game explosion that overshadowed the second half of the 4th console generation.
• Super Mario World - The best-selling, most frequently bundled SNES game, which also stands in for the platforming genre that many of the other best-selling games on the SNES were a part of.

N64

• Goldeneye 007 - Stands in for the FPS genre, Rareware's extensive N64 library, and in part the emphasis on multiplayer for up to 4 people that dominated the N64 zeitgeist.
• Mario Kart 64 - The N64's library was something like 14% racing games.
• Pokemon Stadium - Pokemon was the second wind for the N64, with the Stadium trilogy, Snap, Hey You Pikachu
• Super Mario 64 - Represents the 3D platformer and action-adventure game

Sorry about the N64, but thinking through and writing these out actually takes a little while.