The main console games I play are:
GBA: Mother 3 - the recent fan translation.
An amazing game, that was no doubt a huge inspiration for undertale. It has never been officially translated into English, or been commercially available, so this is probably the best way to experience it portably.
PSX: Tales of Phantasia - English patch.
This was originally a SNES game that was translated into English, while maintaining the Japanese voice acting. Later on there was a GBA remake, however the English voice acting was terrible, and the music was just all wrong! All was fixed with superior updates sprites and a 3D overworld in the PSX remake.
N64: Banjo Kazooie.
I see you mention that N64 games can’t be played well. Have a look at some of the recent development re: Lima driver updates. (Eg the custom DEOT image I made) Banjo Kazooie actually runs really really well on the gameshell! I would have said Zelda, but I guess I’d rather play that on my 3DS. Banjo Kazooie portable is a dream come true! It just has a real charm to it that other games simply can’t match.
SNES: Breath of Fire 2.
This has technically been remade for the GBA, but like I mentioned for ToP, the GBA ports just don’t have the same amazing synthesiser patches the SNES has. Not to mention, SNES emulation is much MUCH faster. I would mention Final Fantasy games, however they have been re-released on other portable consoles, and improved upon. Breath of Fire 2 is a real gem that can be overlooked, and really suited to the gameshell console.
NES: Castlevania II: Simon’s Quest.
Before there was Dark Souls, there was Castlevania. This game makes you back track like a mad man so much, that playing it on a device that doesn’t allow state saving/fast forwarding would drive you insane. And playing it emulated on a hi res computer will leave you wondering what you’re doing with your life. Enter the game shell. The low res screen actually does NES games a real favour, quasi interpolating the image, giving the closest thing to a CRT feel, without lugging 30Kg of wood and glass around. Only portably, could I ever consider playing this game in its entirety.
GB: Kirby’s Dreamland 2.
A throw back to my childhood was using my SNES super game boy adaptor. This allowed some games to have extra thematic borders, added DSP sound effects, and an extended colour palette. Kirby 2 did an amazing job of this, with dynamically changing borders and palettes depending on which world you were in, and whirring wind noises while travelling by warp star. Not too much of a spoiler but let’s say you get to see a finale including a rainbow. Due to the gameboy’s super low resolution, zooming in on the screen would be a pixelated nightmare. Having it run at a 1:1 scale will give people a horrible postcard like box to play in. This alleviates both problems, again without having to lug a TV or computer around with you.
GBC: Links Awakening DX.
This was the Game of my childhood, so you can imagine how excited I was to play the switch version. There was one element that was left out however: photo mode. A small mini game like experience in the DX version. If you played it on switch, this is a fantastic throw back to where it all began, while using an extensive colour palette and yesteryear charm. Of course even if you haven’t played the remake, this is an amazing standalone game with far more charm and personality than a lot of games today. After you’ve gotten used to the interface, definitely add the next two Zelda GBC (oracle of ages and oracle of seasons) games to your todo list! You won’t regret it!