Ultimately, with the Switch, I'm operating under my thumb rule to err on the side of caution. Throwing around speculation is all well and good, but Nintendo proved quite blatantly with the Wii, 3DS, and Wii U that they took an extremely conservative approach to hardware ceilings. And in all three case (notably with the Wii U) there was a lot of pre-full-reveal optimistic speculation regarding the size of the case, what could fit in, modern hardware, etc. And it turned out that Nintendo went cheap as fuck.
Not to disagree with the general gist of your post, but the bolded isn't really true at all for the Wii U. There's a reason it's still so expensive. Even aside from the GamePad and associated tech, Nintendo made two pretty expensive decisions when designing Wii U:
1. Native backwards compatibility with Wii
2. Large eDRAM pool on-die with the GPU
The former meant they had to use an IBM CPU, which meant they couldn't use a single-die SoC, and instead had to go with a much more expensive MCM. The latter (which was likely driven by BC as well, given the need to replicate the latency of Wii's 1T-SRAM) substantially increased the GPU die size, and prevented them from reducing costs by means of die shrink.
Nintendo spent a lot of money on making sure the Wii U was backwards compatible. Had they ignored BC they could have got more performance at lower cost with a simple ARM+AMD SoC.
GameCube is 15 years old hardware.
There is no reason to believe that a device that operates possibly around 350Gflops in mobile mode couldn't emulate it.
GPU performance is largely irrelevant for emulation. It's mostly a factor of CPU performance (and not generally parallelizable across more than a couple of cores).
Except it does. Simple algebra.
ΔE = P * Δt
Voltage of Li-Ion / Li-Po cells is a fixed material property. 3.3~3.7V depending on chemical cell composition. So the commonly quoted mAh capacities transform to Wh capacities just nicely by multiplying with 3.5 "give or take a few".
A 4000mAh battery at 3.5V is a 14Wh battery. If you want 3 hours out of a 14Wh battery, make sure to draw no more than 4.7W "give or take a few".
As soon as we agree what power dissipation headroom a device the size of the Switch might reasonably afford, we can fill in the variables and calculate.
Sidenote: Wii U is 34W with the major offenders ITO power draw made on a 28nm process. Switch is 16/14nm, yes?
The algebra is relatively simple, but it doesn't give us the battery size without knowing the system power draw, and it doesn't give us the power draw without knowing the battery size, so without any hard information on either it's not really giving us any new info on one or the other.
You can find 256+ GB storage that is cheap, or tiny, but not both yet.
Embedded UFS is available at 256GB size, and likely no more expensive than physically larger SSDs. It would be too expensive for a base model (unless Nintendo wants to bump up the price by about $50 to cover it), but it would be plausible for a more expensive model.