Good question! As you write Vitamin A is quite important vision as it is an integral part of the light sensitive rhodopsin protein. Rhodopsin is necessary for a type of photoreceptor, the rod. It is located in outer segment of the photoreceptor. Vitamin A is also required to other opsins that are needed for the cones to function.
When someone has Vitamin A deficiency it does not mean that they have no Vitamin A. For example instead of having i.e. 100 Vitamin A you only have 30 (arbitrary numbers!).
In the retina, you have the very light sensitive rods and the less light sensitive cones. Rods need a lot of rhodopsin (and hence Vitamin A) as they have to detect small levels of lights (it is thought that a rod can detect a single photon!).
Cones, on the other hand are not so sensitive to light. During the day there are tons of photons*. Since you need less Vitamin A for cones to function, you will see the symptoms of Vitamin A deficiency later compared to rods.
Hence, when you still have a bit of Vitamin A but not enough (e.g. 30 instead of 100) you will first notice it in systems that need Vitamin A the most. It turns out that this are the rods. And yes, if you had 0 Vitamin A you would be blind. But since Vitamin A is needed for many other processes, you will probably be dead at that point.
*Personally, I find it hard to grasp just how large the difference in Illuminance (and hence photon numbers) is for example between day and night. See here for a comparison. As you can see, our visual system has to deal with an enormous dynamical range of illuminance. The fact that we usually don't even think about these differences in light intensities in everday live is testament to how fantastic our visual system is in dynamically adapting to the huge differences in input.