Can Pokemon Bank/Transfer be downloaded onto a factory reset 3DS at all?

I currently have a 3DS that I ensured was up-to-date with all the applications and updates before the shutdown.

But I'm now thinking about acquiring another 3DS for backup and trading purposes, and I'm wondering if it'll let me link my Nintendo account at all, and if I can download updates/applications that are already registered under my Nintendo account (Pokemon Bank etc).

Reason being if I get this other 3DS and its only functionality is whatever it has out of the box, I might as well not bother cuz I have no experience in modding.

Edit: thanks guys for the answers. Completely forgot I couldn't link my NNID to two separate devices, but it is reassuring to hear I can still download updates if I do go ahead and get another 3DS for Pokemon trading (not trying to relive X/Y Lumiose city glitch lmao).

I am slightly worried about the longevity of my current system but I plan on swapping the plates and someday snagging a Majora's Mask edition. I'll give modding a try then since it sounds really simple