1.SNES(Had the mixture of quality and quantity of 3rd Party Support)

2.NES(Nintendo's monopolistic business practices made Nintendo have pretty much all the 3rd Party Support, I don't think the NES games are have as much quality like SNES games do)

3.Switch(Nintendo pretty much has all the indies and JRPGs on their system while getting support from nearly all developers, getting ports of older games and even next gen big hitter ports are common on the Switch now, as Nintendo pretty much has every major 3rd Party Franchise on their system, it's close between the Switch and the gamecube though)

4.Gamecube(The Gamecube had pretty good third party support as it got most of the major 3rd Party Franchises and even some exclusive 3rd Party games like RE and the Sonic games, however it missed out on some big ones like GTA and some sport games like 2K, and it didn't have a huge quantity of games like the other platforms)

5.Wii(The Wii had a lot of 3rd Party support but it was mostly shovelware and don't really consider it legit 3rd Party Support, Nintendo missed out on most major multiplats with the Wii)

6.N64(The N64 was pretty dead with 3rd Party Support, most developers didn't want to put the time, money and resources to compress a game to work on the N64. As a result the N64 had one of the smallest game libraries ever for a major console maker, only 300 games, it was mostly a first party system)

7.Wii U(This one is obvious, no developer wanted to develop games for a system difficult to develop for and that no one brought. The Wii U during it's entire lifecycle went 20 Months without one Physical game released, as nearly no one but Nintendo was supporting it.

NES is objectively the right answer.  It had all of the third party support.  SMS got shut out of support.  Hell, Atari technically had a competing system and was still releasing games on the NES via Tengen.  NES really had all of the games.  SNES had good games, but it split third party support with the Genesis with TG16 getting a bit as well.  Anyway, it's all a matter of taste, and most people didn't really experience the NES era.  Here is my list with some of my favorite third party games:

1. NES (Ultima 4, Mega Man 2, Castlevania 1&3, Blaster Master, Final Fantasy, Dragon Quest 1&2, Duck Tales, Life Force, Contra, Double Dragon 2, Gauntlet, TMNT2, Bionic Commando, Ninja Gaiden 1&2)
2. Switch (Puyo Puyo Tetris, Octopath Traveler, DQ Builders, Axiom Verge, Untitled Goose Game, Wargroove, Cuphead)
3. SNES (Final Fantasy 4&6, Actraiser, Super Castlevania 4, Turtles in Time)
4. Wii (mostly Virtual Console, and some ok 3rd party titles)
5. Wii U (Minecraft)
6. Gamecube (basically nothing)

Handheld rankings are a little harder, because my #1 & #2 are close:

1. Switch (see above)
2. GBA (Astroy Boy, FF Tactics Advance, Castlevania: Order of Ecclesia, lots of great ports)
3. DS (Final Fantasy Tactics A2, Plants vs Zombies)
4. 3DS (Theatrhythm FF)

1) Color TV-Game
2) Virtual Boy
3) Wii U

Wait I'm doing this wrong

Home Console:

- SNES
- Switch
- NES
- GC
- N64
- Wii U

Portable

- GB/GBC
- GBA
- 3DS and Switch Tied
- NDS


Being from Europe changes this list as a number of games released never made it over here which is why NES is third for home consoles.

For my personal tastes...

1. SNES 

From Contra III to Demon's Crest to R-Type III, just so many awesome titles 

2. Wii

For all that it was maligned, I own perhaps more third party games for it that any other Nintendo system; games like Monster Hunter Tri, Silent Hill Shattered Memories, Red Steel 2, de Blob 1 & 2, and Goldeneye 007 gave me many long hours of fun.

3. Switch 

It's third party catalog includes some of my top ten games of the past decade, and it looks like this will continue into the 2020s.

4. Wii U/Gamecube 

Both have relatively few games of interest to me, but among those few are some really cool ones like RE4, Black Ops II, and Sonic Racing Transformed.

5. N64

Very little of interest to me; this system was all about Nintendo + Rare in my book.

6. NES 

Famous though many of its games are, I just find games of this era very hard to enjoy for the most part.

1. NES: I actually hated Nintendo first party games with very few exceptions. They always felt basic and cheap aside from Punch Out, SMB2 and SMB3, and maybe a couple of others. I was there for Konami, Capcom, Sunsoft, etc!

3. SNES: Nintendo stepped it up big time, imo. But then so did 3rd parties. This time, Nintendo was the main event but the support was right up there making legendary hits.

3. Switch: Shocking to me as well but my 3rd party Switch games take up just as much of my playtime as first. Almost every game I want ported gets ported.

4. Nintendo Gameboy: The support was there, I guess. But it was mostly shovelware. I had about 13 games but nothing mind-blowing that I can recall.

5. You know what? Nintendo released a lot of hardware! I'll just say DS, GBA, GameCube, Wii, N64, Wiii U, Virtual Boy in that order.

Just going to put in Gamecube for #1. Which I'm sure will be a controversial pick.

The thing is while the NES, SNES, and probably even N64 has way better support in terms of volume, I feel like most games from that era haven't aged that well. Aside from the Mega Man games, I really can't remember the last time I replayed an NES third party title for instance. On SNES, Mega Man X and Square's RPGs are the only games I may replay. And on N64 just the wrestling games.

Gamecube had much fewer third party titles, but there are titles I routinely go back to. Tales of Symphonia, Resident Evil 4, Soul Calibur 2, WWE Day of Reckoning, Capcom vs SNK 2 (wish it had an online mode :(), Skies of Arcadia Legends, Metal Gear Solid, and a few more are games I actually would and have replayed recently. It's mainly a function of games of that era having aged well (beyond the simplicity of early games, but mostly past the awkwardness of early 3D) but there's just most stuff I still enjoy.

1. SNES
2. GBA
3. NDS
4. 3DS
5. Gamecube
6. GameBoy
7. N64
8. Wii
9. NSW
10. WiiU

I mean, do I even need to explain myself? During the pre-Playstation Nintendo era, they had basically everything worthwhile from third parties, on their platforms, mostly exclusively.

Ok, so looking at the ones I've owned, and in relation to the amount of classics, hidden gems and general amount of games:

1st. DS. By far the strongest 3rd party library out of all of the Nintendo systems I've owned, not only in terms of quality but also in quantity. GTA Chinatown, Etrian Odyssey, the definitive version of Chronno Trigger, the Castlevania DS games, TWEWY... You can find dozens of hidden gems easily enough, especially if you like JRPGs.
2nd. Switch. Best 3rd party lineup thanks to being the default port machine of this generation. Most PS3/360 games are going to end up ported there eventually. What holds it back is its lack of more current games, a library needs more than ports to shine.
3rd. GBA. Great solid lineup of 3rd parties. Very similar to the DS only it has a more limited amount of games. Still, it gave us games like the Megaman Zero series, Sonic Advance series, great ports of Final Fantasy I-VI (minus III for some reason)...
4th. 3DS. Great library overall, but it lacks in terms of heavy hitters. Other than a couple of big 3rd party titles like Bravely Default or the Monster Hunter games, most 3rd party titles are either budget titles or niche JRPGs, which, while ok, is not enough to make it stand out.
5th. Wii. The Wii has one of the largest libraries of any Nintendo system, which grants it a lot of good games, like No More Heroes I-II, Sonic Colours, Muramasa TDB... However, the sheer amount of shovelware bloats the library, and makes finding hidden gems rather difficult. The DS has a similar problem, but it ended up having better games overall, maybe because the Wii gave many devs a "follow the leader" mentality and just added motion controls, whereas the DS's gimmick was easier to implement. Also, the Wii is THE definitive version of Just Dance machine, with only the Switch being close to it, so take that as you will.
6th. GameCube (because the Wii is BC). It has a decent amount of great 3rd party games, but outside of them, there are very few hidden gems.
7th. GB (because the GBA is BC). In terms of 3rd party games, only Tetris comes to mind. Other than that, most Game Boy games are either shovelware, or bad versions of NES/SNES/Arcade games.

1. SNES had the best quality plus great quantity
2. NES. It was basically the only game in town, so third parties had no choice but to develop for it. In terms of quantity it can't be beat, though technical limitations hurt the quality.
3. Gameboy. Similar to the NES.
4. GBA. Great in both quality and quantity, but too many of its best third party games were SNES ports in my opinion.
5. DS. Very good argument for it ranking higher than the other handhelds.
6. Switch. Not perfect and many older ports, but it's still shocking how good third party support has been and what franchises we never thought we'd see on a Nintendo system have been coming.
7. Wii. I think the third party support was great for the Wii. Capcom, Ubisoft, EA, Activision all released a lot of great games for the Wii. I have more 3rd party games for Wii than for any other system, though Switch is going to catch up soon. Instead of looking at what it doesn't have (Bioshock, Mass Effect, Elder Scrolls), look at what it does have.
9. Gamecube. Has a reputation for multiplats being worse on it because of issues with the smaller disks, but a good library all the same. As the generation went on its support seemed to get worse even with RE4 towards the end. Very close to the Wii overall I think.
8. 3DS. It received a lot of support because it was the only successful handheld its generation, but it feels like first party games propelled it more than with previous Nintendo handhelds. Capcom was really excellent with RE Revelations and the Monster Hunters.
9. Wii U. It gets the edge over N64 because of the indie scene bolstering its library. But with its early glut of inferior ports and latter desert of major third party games it's at the bottom.
10. N64. It's not like it had nothing, but this was the gen third parties jumped ship the most. The lack of third parties caused huge gaps in the library, especially of RPGs, that no other Nintendo system has suffered from. Paper Mario came out at the very end of the generation and there was basically nothing else.