Yes it's possible but Nintendo would have to make some major, major changes in how they view hardware and how they compete in the general marketplace.

If it's another "hey we're Nintendo, we're going to make a weird console" ... then no it won't get a lot of third party support unless it has a controller that sells to soccer moms/dogs/cats/some other audience that won't buy a console.

It must be a forward thinking platform of some kind too. Simply releasing a "Nintendo PS4" mid-cycle even if it is easy to develop for or whatever will not work either. It won't have the userbase the PS4/XB1 have so it can't be a priority for developers and it will be quickly outdated by the PS5/XB1 (same thing that happened to the Dreamcast, though the Dreamcast was actually far more powerful than the N64/PS1). 

Nintendo needs to figure out an answer to the mid-cycle Dreamcast problem. 

It doesn't matter how well the NX sells, Western third parties will only support it at launch and then will drop all support for reasons. If it does sell well, companies in the East like Capcom will continue their support like they have done with the 3DS.

$$$ = companies

I guess. If the NX is what people are claiming it to be then Nintendo should at least be aiming to get Japanese devs to reconsider releasing their big games on Xbox and instead release them on NX. One would think the games would sell on Nintendo platforms if Nintendo got rid of that casual only console stigma. Just saying.

Yes. If Nintendo really, really wanted to get decent third party support, they'd just have to invest money into it. Take some of the risk from publishers. Cover some of the porting costs. Do that and if you also have an x86 architecture, I think they could eventually get decent sales for 3rd party games, and they would no longer need to help 3rd parties, because they will have built a sizable audience for 3rd party games on their consoles. Also do marketing deals like MS/Sony...

Yes, the relationship between Nintendo and third parties is mendable, but it requires some effort from Nintendo.

$$$ heals all wounds.

I still think the NX handheld has a really big advantage over other systems in launch, and that is the existence of the Vita. The Nintendo 3DS has been the most popular handheld this gen by far, and even then the PS Vita has a lot of games that aren't on Nintendo's platform. Some of those games would have launched on the 3DS, but the system is too weak. With the NX handheld we can assume that is more powerful than the Vita, so a lot of those games might come to the NX, specially in Japan (and even more so if the system is easy to develop and port to). And because the NX is the only 9th gen handheld, they will have a very tight grasp on the market. Imagine all the games the NX could get in its first years thanks to that (Final Fantasy X-X2, Dragon's Crown, Persona 4, Tales of Hearts R, Dragon Quest Builders...).

It's not so much is the relationship mendable, it's can Nintendo get people who actually buy third party games?

By ceding the entire core console market to basically Sony and MS without a fight, Nintendo's console audience mainly buys Nintendo games, or during the Wii craze, Nintendo games + casual mini-game-athons of dubious quality (that fad has died).

They need to at least target XBox IMO (Sony is too strong right now) and win back some of these consumers otherwise it doesn't matter how lovey-dovey Nintendo and third parties are because this is a business and it won't be a good business for third parties.

I think a bottle of good whiskey and a couple handshakes will have everyone back on good terms.