I think a better question is: Do I care if third parties support the Switch?
Not counting Indie developers (who are great and will generally put their games onto any platform that can run them), meh. Except Skylanders, if there's even going to be a 7th game.
I'm part of the problem, sure - but I haven't bought much for third party on the Wii U (Skylanders aside, Need For Speed was a *lot* of fun. Shame EA had to be dicks about it.) or the 3DS.
I buy Nintendo systems for Nintendo games. Third party games that come along are a bonus, not a necessity.
I agree with this to an extent. The truth is there are plenty of 3rd party games I don't care about. I haven't bought a PlayStation or Xbox to play any 3rd party games that I'm missing out on with the Nintendo consoles I own because the vast majority don't interest me and I've got plenty of games to play and replay on with my Nintendo systems.
However, I'm not opposed to 3rd party offerings and have tried and bought various games ranging across a broad spectrum from Just Dance to Skies of Arcadia: Legends to Rock Band and all other manner of different experiences. Sometimes the game is lackluster like NHL Slapshot Wii and sometimes the games can be one of my favorite experiences on a system like Rhythm Thief for the 3DS. My DS experience would have been a lot poorer not having the 5 Ace Attorney games on it. (If you include Investigations.) One of the few games I was playing with any regularity during my lapsed gamer phase in the middle Wii years was Rock Band 2. The Rock Band craze was a big deal for a while and I'm very glad it was released on a Nintendo console. At it's height, it felt bigger than Wii Sports ever got. It was pretty disappointing when Rock Band 4 skipped the Wii U. That's the kind of 3rd party support I care about.
I'm not interested in first person shooters and most M rated games. So, if Call of Duty, Battlefield, Titanfall and whatever other big shooters stay with the PlayStation and Xbox systems, that's fine by me. I don't see them ever becoming popular enough on a Nintendo console anyways. In the past 15 years, PS and Xbox have cultivated that type of gameplay and style and made it a part of their brand identity just like Nintendo's identity is bright colorful platformers with Mario/Kirby or adventure games with Zelda/Metroid. PS and Xbox have built an online infrastructure and culture to support those games and that style and it's one I'm not interested in.
If you remove all of that, what are you left with? Sports games? Sure, it'd be nice to see those on a Nintendo console. I don't see why they shouldn't appear on one but I think that ties in a bit to online play and communities and Nintendo's online policies / capabilities are different from the competition to prevent the user base for those games from wanting to make a switch from PS/XB to Nintendo for those titles. I personally don't care that much about sports games so, again, I'm not that bothered if they don't appear on a Nintendo console but I know others who would like them and, image-wise, I don't think it looks great to a consumer that something with a broad appeal like sports games aren't available on a Nintendo console.
I find the vast majority of third party stuff I like are usually games that are/were exclusive to Nintendo consoles or fit the Nintendo brand better. I know people criticized Ubisoft when they made a statement about determining which games they think would sell on Wii U like Just Dance instead of releasing something like Watch Dogs or Zombie U but I think they're right. The Harvest Moon series fits in well with this idea and Doom does not. Rock Band and Guitar Hero with their peripherals and gameplay matched ideas like DK Bongo Drums and Microphone games. Plus, those games offered pretty much the same experiences as the games appearing on the competition's consoles. Grand Theft Auto: Chinatown Wars, despite being a good game according to those who played it, did not sell well. It wasn't what fans of the series from Xbox and PlayStation wanted nor was it what Nintendo fans wanted with a GTA game finally appearing on a Nintendo system. I think that's a factor why some of third party games sell well and some do not and why we some third party games are still being released on Nintendo systems and some are not.
I'd be happy with GC level third party support but what I'd love to see is DS level support. You may not have gotten games like Assassin's Creed or Destiny on the DS but you did get a lot of new and exclusive titles and experiences. The DS leading into the 3DS started becoming an RPG powerhouse. The World Ends With You, Radiant Historia, Dragon Quest IX. We got the Ace Attorney games. We got companies embracing the DS capabilities with Pac-Pix, Trauma Center, Feel the Magic along with other experiments embracing the less hardcore gamers in Sudoku Gridmaster, Korg DS-10, Cooking Mama, and other such ideas. Sadly, third parties don't seem that interested in making different game experiences like this anymore. Too much money tied up in AAA budget games to risk trying a bunch of smaller budget titles. As such, I don't think there will be a huge shift in third party releases as the hardcore will still want their AAA games on the more powerful console unless portability is really that big a factor for people. The knock on the PSP was that the games for it were basically the same as the console version and that didn't help the PSP in its quest to dominate the handheld market. It was the new games and ideas that were a big factor in the DS's success along with courting the casual market by means of those new ideas like Brain Age and Nintendogs. So, I'm just not sure the portability advantage of the Switch is going to be that big of a factor either in making more people buy third party games on a Nintendo console.