Deadly Premonition moved me, and that game had zero cinematics excluding the weird cutscenes.

EDIT: Oh! Metro: Last Light too.

Journey.

I'm going to have to say Metroid Prime and Super Metroid.

Metroid Prime tells you through the scan logs which add to the atmosphere anyway and Super Metroid is told almost entirely through level design and gameplay sequences.

to the moon

I know TLOU is the flavor of the month now, but I didn't expect so many to just focus on that game and ignore all the great games even in just the PS2 generation....

Silent HIll Series: So much of the story was told without reliance on cutscenes. Those scenes played an important part in the story, but memos/notes, locations, and imagery told volumes. It is really the Blade Runner of videogames ... showing us so much but rarely if ever explictely telling us what is going on. It is the job of the viewer/gamer to interpret the story based on visual clues.

Shadow of the Colossus: Sames as Silent HIll, it showed us so much without forcing cutscenes down our throats to explain everything to us.

Deus Ex series: So much to learn from notes/ect. and supporting characters about the world of Deus Ex.

There are others, but those are some of the best examples I can think of moving/powerful stories that did not rely on cinematic cutscenes/presentations.

Alan Wake, Deadly Premonition and Braid.

Have you seen Titanfall?

The Last of Us does alot of the things you mentioned but for characters which you don't meet and events that you were not present. ALot of the story and world is conveyed in game. The cutscene were normally  reserved for interaction between the characters, instances where you would ideally like to see their expression and body language. 

The cinematic approach definitely isn't for all narratives.

METAL GEAR SOLID 4

That's one of the most cinema-heavy games out there.