That is a very wide, very deep culvert.
In fact, it is a river. A very slow-moving river, over three kilometers wide, five stories deep.
A critical factor is what it is lined with. Is it simply excavated out of the existing soil? Is it rock, or clay-like? How sticky is the surrounding regolith?
If this is all 'fair game' for manipulation, then I would suggest that the advanced civilization would have lined it with some form of very smooth, low-friction, extremely durable plasticrete. Give it sides above ground level of, say, three meters to prevent surface soil from drifting in. You make no mention of any winds or storms, or their frequency. Installing 'drift fences' on either side further out from the sides would reduce sediment from blowing regolith. Ideally, the sides would be engineered to produce wind flows that form an air curtain over the top surface, so dirt and dust is completely blown over the top, and not deposited on the surface.
A combination of reducing the sediment before it enters, and sides that prevent it from sticking, would lessen the problem.
Now, put corrugations in the bottom, parallel to the sides, and the sediment is localized into channels. This makes dredging easier. Putting corrugations perpendicular to the sides would produce sediment traps, and dredging would be further localized. Perhaps drag lines, perpendicular to the sides, in these pre-formed channels, would make dredging a routine maintenance procedure. I am thinking perhaps a curved culvert, instead of a flat bottom channel, like half of a pipe, so the silt would naturally fall to a central point along the smooth sides. This would make it much deeper, to maintain the same volume.
But the crutch is the degree of engineering, construction, and materials that you are allowing of this advanced but extinct civilization that built the infrastructure.
A lot of answers here base the flow rate on the slope of the channel. For this scenario, this assumption is not accurate. The flow rate would be based on how much water is removed from the basin. A bathtub has a very shallow gradient, and virtually no flow, until you pull the stopper. Then, the flow rate depends on the size of the discharge drain. This system is essentially a very big and very long elongated bath tub. Apparently it does not drain into an ocean, so the water is removed only for irrigation and consumption. The flow would not be constant, but would depend on demand. The more water is removed, the faster the flow.
Methinks the greatest factor would be the volume of water the basin holds, the amount of water withdrawn, and the amount of water that can be added by the tap (the polar region ice flows). If the tap supplies less water than needed the basin drains. If the tap supplies more water than needed, the basin overflows. If the tap supplies the same volume of water that is removed, the water level in the basin stays level. The gradient is irrelevant. It is the effect of gravity on the entire body of water that matters. Like a bathtub, drain one end and the level in the other end falls with it.
Irrespective of gradient, the more water that is removed, the faster the level falls. The narrower the channel, the faster the water flows towards the drain. It is hampered by the friction with the channel sides, not the slope. Smooth sides, less friction, faster the water flows.
However, the rate of evaporation depends on the flow of the water. The more stagnant the water, the greater the evaporation rate. So a narrower but deeper channel is more advantageous than a wider, shallower channel. A 3200m deep but 45m wide channel is just as effective, and delivers the same amount of water, but much less surface area for evaporation and silt accumulation.