By the same token computers are finite state machines? Yes, technically they are, but it's not really useful to conceptualise them this way.
It's relatively easy to make a model that works essentially the same as the actual integers (most modern programming languages have a built-in or stdlib variant of bignum), while it's prohibitively difficult to do the same with real numbers. And yes, computability is in fact the difference here, because as long as you are able to keep up the illusion of infinite memory, the integer version problem is trivial, but the real version stays just as hard.
-- edit answering the edit: I'm not familiar with that application, but if I'm not mistaken graphical software uses a different model than the one outlined in the OP, based on algebraic approximations. I'm not saying this stuff doesn't exist, obviously. I was talking about the general rationale behind the OP, i.e. using a model like this one instead of floating point computation for generic software.
The issue is one of representation and algorithms. For integers, you can provide a simple representation together with simple algorithms for addition, subtraction, multiplication, division with modulo, printing in decimal form, comparison which one is larger, etc.
For real numbers, we can come up with precise representations as well. Basically ASTs of permitted operations like addition, root of this polynomial, integral of this function, etc. If your set of permitted operations is large enough, you can represent any number that human language can encode. But operations on those representations are non-trivial.
Basically, for reals there is no representational space which is easy for "powerful" operations to navigate. For integers, there is the n-ary representation. But even that representation makes some operations non-trivial. What is the next larger prime number? What are a given integer's prime factors? Switching the representation to represent integers through their prime factors makes answering that question easier, but then suddenly other questions become non-trivial like addition of two numbers.
None of the two issues has anything to do with cantor.
No computational model can represent all integers, either, by a simple "number of atoms in the universe" argument.
Cantor isn't the problem here.
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Separately, I'm almost certain that exact real arithmetic is being used in Houdini to do boolean intersections. That's as "production" as you can get.