The river touches the edge of the map, but anything I built does not. There's no water being diverted from the river currently, I mean.
The river isn't drying up faster than it can, because it often goes to 2/7 in the 1/7 area, just it is mostly 1/7.

The left side is the edge of the map. It immediately turns to 1/7 where the 3/7 is, so it must be draining. The down ramps were an attempt at trying to drain the water so I could make a bridge to stop it.
The closest to the edge of the map I can add flooggates is where the little rectangle is. That was another failed attempt at diverting water.
Looks like you have a waterfall at the edge of your map, or at least a significant elevation drop. That behavior in flowing water is fairly common.
If you have issues with migrants stupidly crossing the river at that point I'd suggest this:
Your river is 4 tiles wide, so dig an 8-tile wide area underground, smooth the stone at the border of the map, then carve fortifications in it. Then collapse the riverbed so that your river now drains underground off the map.
If you can't drain sufficiently to collapse the river, then cut channels along the side of it several tiles long, and then remove the riverbank by channeling. Here's what I mean:
CRWWWWRC
CRWWWWRC
CRWWWWRC
CRWWWWRC
CRWWWWRC
CRWWWWRC
CRWWWWRC
EEEEEEEEC=deep channel leading to off-map drainage
R=riverbank, to be channeled away
W=water in the river itself
E= last step, connecting the channels on either side of the river with each other so there is zero flow getting to the map end.
Obviously if your river freezes in the winter all this extra engineering is silly. Wait for it to freeze and then chop the river ice and just make a straight shot to the underground drainage.