Will it not allow too much cold air to chill the floor and cause heating problems?
Not if the floor is insulated well. As the others have said the biggest problem is build-up of dampness which can cause rot. There's no free ride here so you end up having to allow for that or risk losing the money saved and more closing it in will bring to making major repairs later on. If you still want a curtain wall then ventilate it very well or you might get the same unwanted experience later on. Modern methods may be better in some ways, but traditional open foundations have been working well for centuries so it's not inherently wrong to follow suit, especially with your house which would have been built that way from the start. Your house, your call.