Yes, but instead of trophies they should have achievements.

I know this is Nintendo so it is much to ask for, but the same goes for the other two: Make it OPTIONAL. I do not want to see these stupid messages "Congratulations, you managed to start the game - have a trophy!" but som people obviously want this and it would not be to hard to let everyone decide if on or off.

I think Nintendo should have a Universal Nintendo Scoreboard, so, for example, you could play the original donkey kong arcade game and compare your scores with those of all people who played that game worlwide.
I like competing for the highest score, in the same way people competed in arcades when I was a kid, but nothing more than that.

I beat the Champion's Road without using Power Ups in Mario 3D World, do I really want a medal for that to show everybody? seems wrong to me.

I'd be more likely to buy one with trophies. I find myself missing them when if play ps1/psp games on my vita.

I don't really care for one... at least not in the way they are right now, but some people really need them so whatever helps with sales I guess... although most of the people I hear complaining about it are people who wouldn't even touch a nintendo console anyways, so who knows

Yes, they should have an achievement system. They can use stars or something.

Nintendo has awards and achievementesque things within certain big games, even Smash Bros Brawl had them. But those are only for your own viewing and they are specific to your game save file not your profile on the console.

Personally I have a slight issue now with games without achievement points, sounds kinda sad but I like having it marked on my profile that I've ranked up every single street fighter 4 character in online fighting.... its better than me saying that I could rock someones world with Ganon in smash melee... I've no proof I could only if I could play against someone in offline game on the GC. Wow that got diverted....  tl:dr Give achievements.

No, I get the feeling you haven't played too many Nintendo games if you are complaining about replay-ability.

Take Zelda for instance, after you beat the game once, there are always things like hidden dungeons which have super hard monsters for you to fight, or you can go around the world collecting heart pieces and completing mini-games.

Now take Mario, you only need 70 stars to beat the game, so you can keep going until you get all 120. Unlock all the secret power-ups etc.

On to Kirby's Epic Yarn where you can easily finish the game, but actually getting 100% completion on a level is a downright nightmare. One of the most difficult games in Nintendo's library if you go by that.

Super Smash Bros, if playing it with your friends all day and completing story mode isn't enough for you, there are hundreds of collectables which appear randomly everytime you play the game. Lots of minigames as well.

You see, those are meaningful things.

Yep this ^. I have the same issue when playing psp/ps1 games on my vita. I also tend to prefer playing trophy based games over non-trophy games if I get the choice. 

trophies only add replay value when you can't afford many new games to play. So doesn't bother me.