Aren't you really just talking about the level of abstraction being applied? If there are infinite possibilities between zero and one, then it's just our way of relating to it that imposes the perception of a limitation.
Now, on logistical level, the mapping procedure is not free, it does have a physical/energetic/mental cost to handle it, and a more trivial representation of integer has a lower cost on this regard.
But the simpler integer representation can represent less many situations. If all one need to deal with in a certain frame is covered by a simpler shipper representation, from an economical perspective it can be argued it worth the risk of hitting a point where the representation will be insufficient as the situation evolved with cases it can't handle.
My impression is that your underlying premises are conflating all of this distinguishable aspects. But of course my impression is limited by my own narrow representation of the world, no matter how open and curious I can be about learning from experiences it makes me go through.
I sometimes wonder how much we allow the map to become the territory.
That said, I accept that literally, 3 never occurs between the ∞ of 0-1 -- unless, we are assuming symbolically, rather than actual, that in our particular ∞, something does not transcend entirely, which we really cannot know, because the closest we have ever come to a real infinity is a representation or symbol of it.
Edit: I realized that the question appears, or is, stupid. But when I see 0.3, I tend to see the 0.0, 0.1, 0.2 between. This is not what 0.3 is supposed to mean though. But on the testability of any real infinity, I do not budge.
on edit: maybe should say infinite set instead of infinite sequence.
What leads you to claim that you can't encode infinite information given an infinite set to work with?
Infinity is an abstraction that simplifies a lot of maths, it isn't real though.
If spacetime is infinite in extent yet discrete at the Plank scale, it would be ℵ₀; the Hilbert space is real-valued and therefore (a vector of) ℵ₁.
The Bekenstein bound, in practice, would do limit how much information can be actually encoded, even if all quantum states have an underlying real value and therefore even just one quantum value of one particle is ℵ₁ all by itself.
But cardinality of real numbers might not be ℵ₁. It's literally unprovable in ZFC.
I don't have a formal education in QP so this comes from my engineering perspective of working on quantum computers.