I don't think anyone's talking about winning the generation, really. It's more about having a sensibly-priced console. The main problem with the PS3 was that it was, what, $200 more than the 360? It probably sold better as a Blu-Ray player than a gaming console, initially (since standalone Blu-Ray players were often the same price as the PS3 or more). However, once the price fell, the sales quickly picked up. It didn't lack the hardware power like the One does, the two consoles were pretty equal in performance (though the PS3 took more time to master, as far as development was concerned).
Here, we're talking about an Xbox One whose price is too high, but it's paired with obviously-inferior hardware. The Kinect isn't Blu-Ray, some widely-useful feature beyond the console it is made for, so it's harder to justify a higher price tag for the Kinect than the inclusion of Blu-Ray for Sony. The two scenarios aren't as similar as you might make them to be.
If it wasn't for the Blu-Ray, I don't think PS3 would've ever caught up with 360, so you are right, it most certainly was one thing that attracted customers.
At the same time, should Microsoft feel it's falling too much behind in sales, it can probably slash a decent chunk off the price and not even be in trouble due to the deep pockets of MS. Sony on the other hand wouldn't really be able to afford that, cos the company is basically losing money on every front except on PS4 at the moment. At the same time, outside of Kinect, the internals of X1 are likely a decent chunk cheaper. Console's become cheaper over time not only cos the hardware gets old, but they learn how to make the "old hardware" more efficiently and cheaper. GDDR5 isn't exactly cheap, while DDR3 is dirt cheap as it is. A more powerful GPU is likely more expensive than a less powerful GPU too.
Then there's the somewhat unlikely option of dropping Kinect, that would considerably drop the price without making them lose profit per console, but that would likely be a very last resort and I think they rather take loss per console with Kinect than profit with console without Kinect.
I don't think we're seeing either thing happen just yet cos MS and Xbox One is doing quite the opposite of badly. Sure, PS4 sells more but at the same time you still can't ignore the massive difference in available markets and even then, even a few million units more sold for the PS4 would hardly mean tha Xbox One is screwed in any shape or form.
As for lacking hardware power.. on what standards? I do get the argument if you sort of make it relative to price. Then again for me (and many others) the price difference and HW power is offset by it's other capabilities. Even if there was no Xbox One and PS4 would be 300$, I doubt I would've ever picked it up personally. You often hear about these magical "next gen standards" that have been on PC for a decade while at the same time even the PS4 can't reliably reach the magical 1080p/60fps in half of the AAA-titles.
But yea, the situation is different, but also similar. At the same time MS can afford doing a lot of things if it starts to look grim for the Xbox One, Sony can't. However, it doesn't look even remotely grim for either console. Only console it looks grim for is Wii U
EDIT: It would be really interesting to know how things would've panned out if Sony had revealed their device first (and not have the option of changing anything after they saw reactions to X1) and have X1 come second and allow MS to react. That said, "sensibly priced" device, I don't think X1 is overpriced, I think it's more about PS4 having a REALLY competetive price. Considering around here we pay 600€ for flagship smartphones (or in some cases even more, fe. Apple), 500€ for the X1 seeing it's capabilities seems like quite the bargain to me, but I guess again it's all relative.