That's it. Xbox One could drop to 399 or even 349 provided they're willing to drop Kinect, which I'm betting MS suits are getting ready to do right now (probably at E3 orf course) in view of weak sales.

So lets say a $349, no Kinect, Xbox One existed. The question is, could it outsell PS4? Just in North America/UK or worldwide? Or is the most powerful hardware impossible to overcome? Would people still prefer the $399 PS4 for more 1080P games?

I think that's the question MS has to be scared of, if the answer is no. Then, like Wii U, they truly are trapped.

349.99 No, PS4 is still a superior product that costs a bit more.

299.99 would make it a hell of a brawl in NA, Sonyland i mean main land Europe would still favor the PS4.

A $349 SKU will easily outsell PS4 in US and UK and with a good margin. They are already head to head now. That is enough for them I think. And let's be real. The wiiU path is very different, X1 sales were never bad in absolute terms like the wiiU one have been for a long time. They are going to surpass the wiiU later this year despite releasing later.

Did being cheaper and more powerful help it last gen?, worldwide not really.
So being cheaper and less powerful this gen wont, either.

Wasn't the X1 cheaper in US than the PS4 this month? If you take into account the free game it was. And look at the giant push it got from that.

Even if it was (and I don't think it was) it doesn't matter. It's the entry price that counts.

I don't think being cheaper will really help the X1 too much at this point. The PS4 has so much hype behind it that I think a price cut would be null by it. They need some killer games like Halo for example to help them. Games are the answer and the best place to address that will be at E3 this year. They need a strong E3 if they want to steal the momentum back. Also, the X1 isn't doing bad—it's doing really well—the PS4 is just beasting it.

Its the value a consumer perceives that counts. 

Even at 350 fo a X1, the PS4 is a more attractive buy. 

In the Global there is NO WAY Xbone could outsell PS4. Also, if you drop Kinect, in the end what do you have ?! A weaker piece of hardware without Kinect. How can you compete with PS4 ? With Kinect al least you can say hey 'we have Kinect included', we might spend 100$ more, for the ones interested. Without Kinect ? What do you have ? Why you should buy an Xbone ? I mean, apart from Halo and Forza hot fans. With 360 was different, multiplatform games were better for the majority, it was cheaper, great graphics, and better shooters. Now, also shooters will be better on PS4 (as any other multiplatform game), Call of Duty, Battlefield, Destiny, etc, will be better on PS4.
MS could just cut the price of the whole package, or release the box without Kinect. Whatever you do, they will lose money for a temporary boost, and down again, below PS4. MS did a BIG mistake with this One, most of all considering it is coming after the 'perfect' 360.

Don't think so. At this stage the entry price counts more than the value. You can put how much you want in a bundle, if a person earns that much there is only so much he or she can spend on videogames. $500 is way too high now. People will start considering more the value when we enter the 199-300 price range.