I'd suggest it's actually quite realistic.
If we look at the history of this land mass, we might have seen something like this a short while ago. Triangles indicate "high" land while the multi-peak icons represent actual mountains. The white and grey dinosaur skull represents ice.
Here we see the continent as it was during the most recent glaciation. Ice being quite heavy, it has pressed down on the land, especially in the areas marked light grey, where accumulations were densest. Melting along the edges is occurring and is drained away mostly to the south, where four archaeorivers have cut through the uplifted Southern Highlands and also in the west where a gap between the mountain ranges allows for outflow.
Moving up to the present, post-melt period, we might see this:
We have the Great Central Spine, which I'd imagine represents the collision of two smaller proto-continents and also the Great Southwestern Mountains, which might represent a lesser collision or collision in progress of a plate from off the map somewhere.
The ice now melted, what we have left is a ring of higher land, pushed up, and two central areas of depressed land, pushed down, due to isostatic forces. The grey areas where the ice was most concentrated and thus heavier and more oppressive formed deeper basins within the two great basins, and these are where residual melt water that didn't escape Agassizwise through the western channel now forms a series of endorheic lake & river systems.
I would suspect that in some distant future, as the central lowlands continue to rebound from their former depression, the endorheic lakes will disappear along with the ring of (relatively) high land, most of the rivers will reverse course and the lakes will drain to the surrounding Ocean.
Your scenario reminds me a lot of this land: