I know dolphin, at least, is subject to the same shader stutter that emulators for newer platforms are (if I understand correctly it's a shader per set of TEV parameters, or something like that), with the only real mitigation being that they've been able to write an "ubershader" that can (slowly) emulate the whole TEV pipeline in software for impacted pixels while they generate and compile an optimized shader in the background. Obviously Nintendo would probably ship shader caches to address the problem of having to generate them on the fly, but apparently the reason why dolphin has to do this is because games can just reconfigure the TEV with no warning and get results basically instantaneously. That definitely sounds like dynamism to me, but I don't know how it maps to this new extension.This is very interesting indeed. I don't think it pertains to new hardware at all (ray tracing not being supported yet would certainly be a limiting factor), but it's very interesting not just that Nintendo is contributing to Vulkan directly, but they're contributing a major feature which fundamentally changes one of the key paradigms the API is built on.
I'm definitely no expert on graphics APIs either, but I would be surprised if this relates to emulation or BC in any way. From an emulation perspective, it would absolutely make sense that it could allow the dynamism supported by the graphics API being emulated to be better mapped to Vulkan. However, the only prior Nintendo hardware which would seem to warrant this would be the Wii U, as it's the only console other than the Switch which supports programmable shaders. In my understanding (which may absolutely be wrong), emulation of fixed-function hardware like GC or Wii would map well to the existing pipeline paradigm. And for reasons that should be obvious, I would expect Wii U emulation to be very far down Nintendo's priority list.
In terms of BC, I can't see any good reason to involve Vulkan in Nintendo's BC efforts. The benefit of Vulkan is that it is an open, general purpose, cross-platform API, and the benefit of contributing to it is that you can then depend on that contribution being supported across a variety of hardware and driver implementations. The implementation of backwards compatibility is almost the opposite of this; it is a very specific problem, for which a single implementation is made, to run on a single target architecture, and the entire stack can be (and probably is) proprietary. Nintendo and Nvidia can implement this in any way they wish without any regard for anyone else, so why would they constrain themselves by implementing it on top of a cross-platform API?
My guess is that Nintendo's involvement in Vulkan, and their usage of it in emulators, is a kind of future-proofing or insurance for future hardware. In the case of the emulators, there is probably little to no benefit from using NVN over Vulkan, but if they ever change hardware vendor in the future, then using Vulkan can at least mean avoiding having to re-implement their emulators as they had to do moving from Wii U to Switch. It's very unlikely they'll actually use Vulkan for games, but if they do change hardware vendor, they will have to work with them to implement a new graphics API for the new hardware. Using Vulkan as the starting point for such an API would make a lot of sense, as it's an API Nintendo is already familiar with, and any hardware vendor will also be familiar with and have a mature driver implementation for. Proposing improvements to Vulkan like the one we see here could simply be a way for Nintendo to push it closer to their "ideal" graphics API, and the closer it is to that, the less friction there would be in implementing any new custom graphics API in the future.
As for why they might choose to use Vulkan for their BC layer, I think future proofing may be a compelling concern there, as well. I doubt that they'd go with a fully generic solution for performance reasons (in particular, the shader translator itself seems likely to be highly specialized), but keeping it as generic as possible while meeting their performance goals means less work bringing it forward beyond Drake. As time goes on, they could just Ship of Theseus it into a full on emulator the more that their hardware drifts away from the TX1. As proposed by @LiC, there's a certain appealing logic to using custom, highly hardware specific APIs like NVN in the moment to make games, while using more open standards like Vulkan to preserve them. Nintendo's got a good thing going with Nvidia right now, but they also had a good thing going on with IBM/AMD until one day it wasn't going so good anymore, and as you pointed out, they had to eat a fairly costly transition. In this current environment of much more expected BC, designing with an eye towards the longer term is probably a wise move.