A lot of the frustration in these early games comes down to no save function, which is fixed by the emulators. 

So pick a game that's meant to be good and just save-cheese it.

Otherwise, the early Zelda games are great and hold up well.

Unfamiliar with the device, but here's some from the top of my head and from my experience that seem less on the masochist side:

Neo Geo: Metal Slug X, 3 and 5, I'd say.

SNES: Final Fantasy VI

Mega Drive: Tanglewood (and yes, it's an actually modern game)

GBC/GBA: maybe Shantae? It feels in an odd spot between arcade difficulty and streamlined/story-oriented difficulty.

Apple IIGS & Macintosh Classic: the MacVenture series.

NES: Nyx: The Paradox Relic (also modern game)

For that device I personally would focus more on SNES and GameBoy gaming (up to Advance), so I'll just mention a couple of my favourites.

SNES:

• Chrono Trigger > One of the best JRPGs ever. If never played before, well worth giving this a try.
• Super Mario RPG Legend of the Seven Stars > Sooo much fun, easy to burn hours.

GameBoy:

• Pokemon Red/Blue/Green > Original Pokemon series, if you love it you love it.
• Pokemon Yellow > The Special Pikachu Edition has Pikachu following you around, its cute, and the one I would recommend personally out of the originals.
• WarioLand 2 > One of my favourite platformers on the GB.
• GameBoy Gallery 2 > Contains old Game and Watch games but with a (at the time) modern take on them + the original versions. I've wasted hours beyond hours playing modern Fire.

GameBoy Color:

• Pokemon Gold/Silver > It's Pokemon lol
• Swiv > Kind of forgotten title, but really fun scrolling shooter (I think this the best way to describe it)

GameBoy Advance:

• Pokemon Fire Red > An updated version of the original Pokemon Red. If it's your first time playing any of the originals, I would say try this one as it feels more modern to play due to the updated pixel art.

It looks like the most modern retro console it can handle is the GBA. Personally, that leaves you a bit limited as I personally feel anything older struggles to have a modern feel (not that there aren't good games, but I think it becomes a lot more about personal preference/taste on what counts as modern).

For the GBA any first party Nintendo games will hold up pretty well. I think Advance Wars, Metroid, Zelda games would even be hits if released today.

The Megaman series is actually a pretty good one. Megaman Zero series is good 2D action, and the Megaman Battle Network games have a fascinating grid/deck building gameplay that was also used in One Step From Eden.

The castlevania series on the GBA is also one that would feel at home if you like metroidvanias.

Admittedly those are all more action games, if you don't mind a more traditional RPG or puzzle game there are more options.

• Castlevania: Symphony of the Night

• Mother 3 (English patch)

• Chrono Trigger

• Mario & Luigi Superstar Saga

• Colin Mcrae Rally 2

• Final Fantasy VI (or Final Fantasy III in the US)

• Apotris (this one is a Tetris homebrew that looks and runs great)

These all feel very modern for their time, and don't come with the hassle of being overly vague or forcing you to grind.

Original Sonics hold up well imo. Plenty of time in them! Also, streets of rage and the tmnt clones are good too. And lemmings for snes I think is a good puzzle game. I struggle with RPGs of the day after Mass Effect, but Pokémon are still solid. I also recommended Advance Wars. And Doom, it plays very nicely on my r36h with the auto-aim mod installed too (tweak it down in assist though). And the Monkey Island series are definitely worth a play through if you missed them first time round

Chrono Trigger, of course. Still feels modern, especially with the resurgence of pixel stuff. Now, I'm not sure it's rotational like something like Mario World would be.

I am having tons of fun with my old PSP. The kids are playing old GBA Pokémon games and I'm replaying link's awakening and minish cap. FF6 and chrono trigger are also in the cards.

Modern/Retro-inspired: Goodboy Galaxy, Annalynn, Cathedral, Alwa

Anything Nintendo: Mario and Zelda, obviously

NES: Mega Man, Crash and the Boys

GB[C]: Kirby, Metroid II, Gun Zas, Solar Striker

SNES: FFVI or Chrono Trigger, Arkanoid Doh it Again, Killer Instinct or MKII, DKC, Mega Man X, Super Metroid!!!!!!!!

GENESIS: Vector Man, Earthworm Jim, Comix Zone, UMKIII

(I missed out on GBA so I don't know...)

Squarez Deluxe is freeware and makes Tetris seem like a child's toy.

E: Squares -> Squarez... Damned auto correct

i found this hack of Arcana to be great, and I don't usually enjoy dungeon crawlers. The auto mapper is a big part of that https://www.romhacking.net/hacks/4505/

Kirby Super Saga for SNES plays a lot like a modern indie platformer (or they play like it I guess)

The SNES remake of Dragon Quest 1 has a simple gameplay loop of walk around a town fighting for money, buy the equipment at that town, go to the next one, repeat that I find very enjoyable. The original game came with a map and guide so don't hesitate to look at a walkthrough.

The miyoo Mini should be able to emulate Playstation, so #inal Fantasy VII (and if you're into those: VIII and IX) are on the table. But they needed modifications to be viable for modern audiences (9999HP + Lvl 99, triple speed, no random encounters, etc.)

Check out the Quintet games (Soul Blazer, Illusion of Gaia, Terranigma) for some relatively unknown classics.

Also check out some mods for the games. Terranigma for example benefits a lot from a rebalancing mod.

If you can run portmaster on it, check out Panzer Paladin.

Via portmaster, sonic triple trouble 16bit.

Luanti might work. i'm not sure how powerful it is. I mostly play mega drive and psx games (penny racers).

The game that gets the most play on my handheld is the first Donkey Kong Country. I think 2 and 3 are also very good. I think they feel modern in the way that trendsetters sometimes do; I think they are still influential for modern platformer designers. And the music is as good as the platforming, maybe better.

I'd recommend the MinUI firmware for the Flip Mini.

As for games maybe Golden Sun for the GBA.

Solomon's Key for NES is timeless.

Arcade:

Bank Panic

Willow

Forest Wardner

Gyruss

Slap Fight

Robocop

Black Tiger

Then check libregamewiki.org for the excellent

Kobo Deluxe

Blob Wars

NeverPutt

Enigma

Fish Fillets NG

Frozen Bubble

Holotz Castle

Shattered Pixel Dungeon

Ardentryst

Search and Rescue II

Mineclonia

Project: Starfighter