Yes, humans like to have variety in their food, and your concern on a game design point of view is valid. A few short ideas about it; some other very clever folks here will probably chip in and give more/better tips but here goes:
Humans like variety in their food because it makes them happy. You could have a "happiness" meter attached to your settlers which includes how much variety there is in their food (I believe Banished has something like this).
Humans need variety in their food because it makes them more healthy. You could have a "health" meter attached to your settlers which includes this as well. Both "happiness" and "health" traits should be tied to the ability/desire of the settler to work and generate new settlers.
Humans need variety in the food they produce because they need to feed animals. For instance, you could feed cabbage to humans and sheep, and use wheat to produce bread for humans and feed hens.
If you have trading game mechanics, allowing the player to sell some types of crops could encourage them to allow more variety in what they plant and grow.
A way to avoid tracking food types once it's produced and still encourage the player to add variety is by adding random events such as droughts, plant illness and parasites. Those events would target only a specific type of plant and kill them before they produce edible food, and so if the player only has that type planted that year, famine arise due to lack of food.
Finally: seasons! Some food needs to be eaten fresh and can't be stored for later, like strawberries, but some food can be kept for colder seasons when crops don't grow (like potatos). Link this to the fact that some plants grow fast and some other grow slow, you make sure that players need to diversify what they plant.
On a final note, I believe Banished, Stardew Valley and Minecraft have interesting game mechanics regarding food and crops, so you might want to take a peak at those games and see how they did it.