Anyone else experiencing this issue?
Can't say that I've noticed it. If anything, I seem to be quicker on the XBone, just because the sticks take less force to move.
My main issue with the XBone is that its party system is idiotic. You can't join someone's fireteam from within the Destiny interface (I mean when YOU are initiating the joining, as opposed to responding to an invite), which you CAN do on the 360 version. On XBone, from within Destiny you can only invite someone else. In order to initiate joining someone, you MUST go through the console's menus. Which would almost be acceptable if they weren't horrendously slow - honestly, it is amazing to me how Microsoft managed to make the interface on their brand new console slower than their old one.
This joining issue is not unique to Destiny, BTW. All XBone games I've played are hampered in some way with regard to parties/in-game groups. This is, I believe, due to the fact that when the console first launched MS insisted on trying to make console parties and game parties the same thing. It was only with a later update that it became possible to have an XBL party separate from your game party. But I think something is still screwy with the underlying architecture.
The only positive thing I can say about the XBone interface is that Microsoft has genuinely been making an effort to address the usability issues. It's much better now than it was at launch, and with Phil Spencer in charge I think it's going to keep improving.