This actually sounds nice. Even my more casual gaming friends still like to tell me of their times at making crazy last shots to kill enemies to desperately earn achievements, or to go over a history of some of the moments involved with earning them.
I generally give a "Woot!" when I earn an achievement that was actually a worthy accomplishment. Such as getting through a no-miss clear on a shooter stage, or beating a fighting game on harder difficulties with no continues. On the "Trophy" side, I felt great when I Platinum'd Gravity Rush + all it's DLCs.
Recording those moments, so you can look back on them, sounds nice. BUT....
...why is anyone complaining about the space it'll take up, or how no one wants to see these images? You can be VERY sure the games will have to ask your permission before uploading anything to a server. It's not like it'll just clog up your game page on the XB1 with the 1000s of achievement videos you earn over the system lifetime... you'll be able to view the content later, and decide if there's a reason for you to upload any of that for sharing yourself.
The fact it's separate from the last 5 min is a good thing, too. If I play a game for 3 hours, and I actually made 1 or 2 worthwhile accomplishments, being able to look at them all, and see if there's any I find memorable, is great.
Did anyone here play Phantasy Star Online? My friend and I used to laugh at the fact that the ending pictures (featuring your characters) were actual in-action screenshots of our gameplay whenever we beat the game. I can imagine XB1 using Game DVR to actually DO this, allowing it to play back your greatest gameplay moments during credits. (Some fighters already did this with death blows in past-gen fighters.)
I could see myself getting some fun out of this over the years...