I'm glad you asked. As you might have guessed, the same question occurred to me in the process of siding with Kronecker and Brouwer on this question instead of with Cantor and most other recent mathematicians.
You are right that "the idea of working with infinitesimals seems paramount in calculus". It is paramount. But you don't necessarily need the concept of infinity, in my opinion. What you need is the concept of a limit so that you can make a reasonable definition of continuity.
Even though a finite math would be "grainy" and not "smooth" like the infinite math we are used to, I think that you could still define continuity in such a way as to prove all the same important theorems. I don't think calculus would lose anything important as a result.
You can visualize it in terms of a video monitor screen. There are a finite number of pixels on the screen but you can still have continuous (sort of) curved lines drawn on the screen. The lines will be seen as continuous if the resolution of the screen is high enough, or if you stand back far enough. Of course, if you get up close, you will see the discrete nature of the pixels. It seems that is exactly what we see in nature when we examine it close up versus examining macro phenomena. So it seems to me that a discrete math would be better suited to describing nature.
A crude definition of continuity, using this video analogy, would be that a line would be defined as continuous if and only if the pixels making up the line are all contiguous.
If you are familiar with the epsilon-delta definition of continuity, then you can see that the role of epsilon, rather than being an arbitrarily small number, would be played by the pixel size of your system, i.e. 1/N where N is the largest integer in your system.
Let me know if this doesn't answer your question, and thanks for asking.