Hypothetically speaking, do you think this would change the gap between PS4 and Xbox One? 

The Kinect was averaging 133,333 sales a day the first 2 months of it's release, breaking sales records. I can't help but think this would have significantly helped the One had the original Kinect not had left a bad taste with gamers. Unfortunately, now, the Kinect doesn't seem like much of a system seller but if the hype from the 2010 release was saved for the Xbox One release, I believe this generation would have started out a lot differently. 

I think it would have been even worse. Microsoft hasn't done much to show off the Kinnect to gamers, and motion controls feel like last gen. It's the lack of Kinnect games that make me resent the added price.

It would have boosted Xbox One's sales, but I'm not sure by how much.

It would have been different ''hype'' of the kinect and next gen = Holy shit

Motion gaming is as dead as Guitar Hero, Tony Hawk games and the Vitality sensor.

Gamers don't want gimmicks and casuals don't want consoles.

But we know we don't care much for motion gaming because of Kinect 1. Had that not of come out, I think a lot of people would have positive hype for it. 

That's hard to say.

PS3 would have had an even bigger momentum at the end of last gen without Kinect reviving/sustaining the 360. Move might have been slightly bigger, Wii motion plus might have had some more attention, motion gaming fatigue would still have followed.
Voice control is not as fresh in 2013 as it was in 2010.
With a $500 price tag, instead of a $150 add-on option, or a cheap 4gb arcade Kinect bundle, the audience Kinect was mainly aimed at wouldn't be as quick to purchase.

The Kinect hype was huge though, it was everywhere on tv. Assuming nobody else picked up the tech before 2013 it would still have felt fresh enough to get people interested.
The effect would have been far smaller as in 2010, but I do think it would have made a difference. Not enough to keep up with ps4 worldwide with the 13 country launch and higher price tag, but XBox One should still have been firmly sold out in NA. (Assuming that the launch Kinect games would be present at the XBox one launch, instead of the current situation)

A lot of ifs and assumptions, 2010 was by far the better time to launch Kinect right after the height of the Wii wave.

It would have been huge and pushed X1 sales significantly. Kinect 1 was the fastest selling electronics device in history and as of Feb 2013 sold 24 million units. I think Kinect 2 could have been much bigger than 1 since its much better and could get some good games.

Yes. We'd have another novelty at or hands and people who usually don't care for the hardcore gaming were going to buy it. But now that the novelty is gone and most people realized that kinect is actually boring and not that well used for games it turned the perception around to even be detrimental to sales.

Yeah it would definately get more hype sales, and some people would still believe it will change the gameplay of core games if kinect on the 360 didnt already disprove that.