All rivers flow downhill overall.
Why isn’t there a deepening canyon? In some cases there is, but rivers are just water flowing downhill on the surface of the land. Rivers generally go the way they go in the first place because that’s the way the land is already sloped, even if it’s so gradual that you wouldn’t think the land wasn’t completely flat. Land is almost never completely flat. Water goes whichever way is down, and where that way is clear enough and there is enough water built up, a stream or river is the result.
Rivers carve out a bit when they are fast enough, but they also carry soil and deposit it, leaving buildups of soil, which can change which way “down” is, resulting in wandering rivers that sometimes change course, leave oxbow lakes, etc.
There are some spots on some rivers where the water goes up very briefly due to momentum having been built up horizontally and then having the lowest way forward be shallower than before, but that will erode away, and is hardly worth mentioning.