There is nothing wrong with all answers here, but one aspect is missing.
If the nose gear collapses, the instinctive reaction of most pilots would be to brake as much as possible. Forget reverse thrust or spoilers - the pilot will hit the brakes. If pressure builds up slightly differently between left and right brake, or if the wheel load between both main landing gears is different due to cross wind, the aircraft will experience asymmetric braking, and now the nose will not be exactly centered between both main wheels.
Now we need to look at the braking forces of the skidding front fuselage in comparison to those of the main wheels. Since the aircraft has now a pronounced nose-down attitude, its load on the nose will be quite a bit higher than the regular load on the nose landing gear used to be. With the increased load on the front fuselage, its scratching along the concrete will contribute a lot to the braking force, and if this force is off-center it will produce a yawing moment that can be balanced only initially and with full application of differential braking. Once the front fuselage has sverved beyond the tracks of the main gear, no braking will bring it back into the center and the aircraft is uncontrollable. Add some crosswind and it will leave the runway quickly.
Is it possible to still control the aircraft during this weird event?
Applying rudder will still help at high speed, but once the aircraft slows down and the rudder becomes less effective, the aircraft is uncontrollable.
Or is it more or less inevitable that the aircraft will run off the
Only in perfectly symmetrical conditions will it stay on the runway. Since the aircraft with the collapsed nose gear is unstable in yaw, the far more likely outcome is a runway excursion.