Yes, your answer is fundamentally wrong. Let me point at that it is not even right in the finite case. In particular, you are using the following false axiom:
If two sets of outcomes are equally large, they are equally probable.
However, this is wrong even if we have just two events. For a somewhat real life example, consider some random variable $X$ which is $1$ if I will get married exactly a year from today and which is $0$ otherwise. Now, clearly the sets $\{1\}$ and $\{0\}$ are equally large, each having one element. However, $0$ is far more likely than $1$, although they are both possible outcomes.
The point here is probability is not defined from cardinality. It is, in fact, a separate definition. The mathematical definition for probability goes something like this:
To discuss probability, we start with a set of possible outcomes. Then, we give a function $\mu$ which takes in a subset of the outcomes and tells us how likely they are.
One puts various conditions on $\mu$ to make sure it makes sense, but nowhere do we link it to cardinality. As an example, in the previous example with outcomes $0$ and $1$ which are not equally likely, one might have $\mu$ defined something like:
$$\mu(\{\})=0$$
$$\mu(\{0\})=\frac{9999}{10000}$$
$$\mu(\{1\})=\frac{1}{10000}$$
$$\mu(\{0,1\})=1$$
which has nothing to do with the portion of the set of outcomes, which would be represented by the function $\mu'(S)=\frac{|S|}2$.
In general, your discussion of cardinality is correct, but it is irrelevant. Moreover, the conclusions you draw are inconsistent. The sets $(0,1)$ and $(0,\frac{1}2]$ and $(\frac{1}2,1)$ are pairwise equally large, so your reasoning says they are equally probable. However, the number was defined to be in $(0,1)$ so we're saying all the probabilities are $1$ - so we're saying that we're certain that the result will be in two disjoint intervals. This never happens, yet your method predicts that it always happens.
On a somewhat different note, but related in the big picture, you talk about "uncountably infinite sets" having the property that any non-trivial interval is also uncountable. This is true of $\mathbb R$, but not all uncountable subsets - like $(-\infty,-1]\cup \{0\} \cup [1,\infty)$ has that the interval $(-1,1)=\{0\}$ which is not uncountably infinite. Worse, not all uncountable sets have an intrinsic notion of ordering - how, for instance, do you order the set of subsets of natural numbers? The problem is not that there's no answer, but that there are many conflicting answers to that.
I think, maybe, the big thing to think about here is that sets really don't have a lot of structure. Mathematicians add more structure to sets, like probability measures $\mu$ or orders, and these fundamentally change their nature. Though bare sets have counterintuitive results with sets containing equally large copies of themselves, these don't necessarily translate when more structure is added.
Let's completely forget all notions of number entirely.
It's not hard to show this operation (which I will call $\oplus$ rather than $\bullet$) has nice algebraic properties: it's commutative, associative, and it respects the ordering on line segments.
Alongside $\oplus$, we could pick some segment to be a unit, and use another geometric construction to define an operation $\otimes$.
These two operations on line segments have very good algebraic properties; without using any notion of number, we've still managed to construct a system in which we can do arithmetic, algebra, and analysis — in fact, once we add in a notion of direction, these operations satisfy the complete ordered field axioms.
What is the length of a line segment? The line segment itself. Addition is literally defined as "extending a length".
I believe this overall description is a relatively accurate description of actual history — e.g. that Greek geometers did algebra with line segments, considering the segments themselves as a notion of quantity; e.g. see Euclid's elements, book 2. The real number system came into being precisely as numeric representations of these geometric objects
In fact, AFAIK, for a long time mathematicians considered numbers quantifying length and numbers quantifying area as completely different kinds of numbers, rather than the point of view today where both kinds of quantities are described by the same number system, but possibly with units attached.
The perspective that treats real numbers as being more fundamental than geometric notions is, I think, mainly an artifact of how math is taught in modern times.
Best Answer
There is nothing wrong with sometimes thinking of real numbers as infinite decimals, and indeed this perspective is useful in some contexts. There are a few reasons that introductory real analysis courses tend to push students to not think of real numbers this way.
First, students are typically already familiar with this perspective on real numbers, but are not familiar with other perspectives that are more useful and natural most of the time in advanced mathematics. So it is not especially necessary to teach students about real numbers as infinite decimals, but it is necessary to teach other perspectives, and to teach students to not exclusively (or even primarily) think about real numbers as infinite decimals.
Second, a major goal of many such courses is to rigorously develop the theory of the real numbers from "first principles" (e.g., in a naive set theory framework). Students who are familiar with real numbers as infinite decimals are almost never familiar with them in a truly rigorous way. For instance, do they really know how to rigorously define how to multiply two infinite decimals? Almost certainly not, and most of them would have a lot of difficulty doing so even if they tried to. It is possible to give a completely rigorous construction of the real numbers as infinite decimals, but it is not particularly easy or enlightening to do so (in comparison with other constructions of the real numbers). In any case, if you are constructing the real numbers rigorously from scratch, that means you need to "forget" everything you already "knew" about real numbers. So students need to be told to not assume facts about real numbers based on whatever naive understanding they might have had previously.
Third, it is misleading to describe infinite decimals as the basic "naive" understanding of the real numbers. It is unfortunately often the main understanding that is taught in grade school, but this emphasis obscures the fact that ultimately the motivation for real numbers is the intuitive idea of measuring non-discrete quantities, such as geometric lengths. When you think about real numbers this way, they are much more closely related to the concept of a "complete ordered field" than they are to the concept of infinite decimals. Ancient mathematicians reasoned about numbers in this way for centuries without the modern decimal notation for them. So actually the idea of representing numbers by infinite decimals is not at all a simple "naive" idea but a complicated and quite clever idea (which has some important subtleties, such as the fact that two different decimal expansions can represent the same number). It's kind of just an accident that nowadays students are taught about this perspective on real numbers long before any others.