Based on the comments on my first answer, I think I can take a stab at this and help out more. There are two issues here. One is the issue of the student's understanding of the need for the concept of limits, and the other is your understanding of the notation that is being used. Your interpretation of $0.\overline 9$ and $\sum\limits_{n=1}^\infty\frac{1}{2^n}$ is not the standard interpretation of those expressions, and that non-standard interpretation is causing additional confusion. Accordingly, I will avoid using those notations until late in the answer, so that we can have agreement.
The key concept that I believe your students are missing is that there is no ultimate (final, last, etc.) element to an infinite series. I believe your students have a sense that there is an ultimate element which is, as you say, "realized." Finite series have an ultimate element, but infinite ones do not.
So the challenge is to convince them of that. Rather than using complex approaches like the Manhattan paradox, stick to the simplest and most famous: Zeno's Paradox. To run a race, you must first reach halfway to your goal. From that point, you must reach halfway to the goal again, and again, and again. Zeno argued that you never reach the goal, because you always have at least one more step in this process. But obviously every person alive believes you can run somewhere, so there is clearly a problem with our thinking: our thinking claims it is impossible to do something which we know to be possible!
If we formalize this paradox, let $d$ be a series which represents how far has been traveled, such that the $n$th element of the series is how far we have traveled after we have reached $n$ halfway points:
$\{\frac{1}{2}, (\frac{1}{2}+\frac{1}{4}), (\frac{1}{2}+\frac{1}{4}+\frac{1}{8}), \ldots\}$
Now, your argument is the same as Zenos. You argue that this "process" can continue forever. If your students believe they can achieve $1$, then ask them which element of the set is equal to $1$. The goal here is to get them to phrase it as something like "the last element."
If they say something like "the infinity-th element" then you will have to work with them to understand that infinity is not a natural number. This is obviously true from set theory, but you may not want to go into set theory. You may have to approach them in an intuitive sense. If the students go down this approach, they have a concept of what infinity is, and how it behaves. This would be a key concept to teach them how to unlearn. I'd query them as to what the result of some expressions are. For example, most people I've worked with who have learned an informal concept of infinity have the idea that $2\infty =\infty$. If they have that assumption, then demonstrating the difficulty of reconciling $\frac{2\infty}{\infty}=\frac{\infty}{\infty}=1$ versus $\frac{2\infty}{\infty}=2\frac{\infty}{\infty}=2$ may be enough to demonstrate that infinity is, indeed, not acting like any number. If your students have a different concept of infinity, you as a teacher will need to learn what that concept is and provide the correction. Perhaps they will be intrigued by the puzzle of whether infinity is odd or even, given that all natural numbers are either odd or even. My expectation is that this will be an easier issue to resolve than the original issue in your question.
So once we get the students to use the phrasing "ultimate element" or something like that, then we can start to cut away at the concept that there is an ultimate element of an infinite series. I think the easiest example is the case we discussed on my other answer, $\{0.9, 0.99, 0.999, 0.999, 0.9999, \ldots\}$. As you have pointed out, this construction consists only of the digits $0$ and $9$. There is no way any element of this series can contain other digits, but if an ultimate element or last element did exist, its very easy to adapt my proof of \$0\overline 9=1$ to show that if you let "S" equal the "last element of that infinite series," my proof works. This creates a contradiction: we can prove that every element in the series consists of only 0's and 9's, but the last element (if you claim one exists) must equal 1. This contradiction will hopefully help the students abandon the idea that a last element of an infinite series exists at all.
I believe this is the intuitive challenge that your students need to overcome. They must become comfortable with the idea that there are series that have no ultimate element. I think this is the root cause of the issue you are trying to correct in your students.
Once you have that, the final step is limits. We all know that we can run places. We know Zeno's claim about the world is false, but it appears to be true about the math. How do we resolve that? The answer, of course, is what you want to teach: limits. The concept of the limit was invented because everyone had the intuitive idea that we can go places, so we should invent a concept which lines up with that intuitive understanding of reality. I'm sure many solutions were invented, but the concept of the limit proved to be the one that made the most sense and was most consistent. Eventually that was formalized with epsilon-delta proofs in 1821, but it was used in varying forms up to that point.
In Newtonian physics, there are implicit limits everywhere. We can easily take the limit of the series $d_n$, which is a finite series whose final element contains the distance traveled after reaching $n$ halfway points, as $n\to \infty$, and see that that limit equals 1. This matches up with our intuition, satisfies the mathematics, and if you look at the mathematics for an object traveling with a constant velocity, and do Riemann integration, you see that it can be visualized as a set of boxes very similar to Zeno's fractions.
So that's the selling point. The selling point is that the intuitive concepts which we try to use to resolve paradoxes like Zeno's paradox depend on either infinity being a number or on there being an ultimate element of an infinite series. But both of those assumptions bring about disastrous contradictions, so both must be false. But we all know we can run somewhere. The solution to this was the invention of the limit, which tied this mathematical problem back to reality.
Now, you will note that I explicitly wrote out every sequence in that explanation. I didn't rely on any shorthands. When you want to introduce notations like $\sum\limits_{n=1}^\infty\frac{1}{2^n}$, this is where you will need to be precise, or you will confuse the daylights out of your students.
Take the summation symbol, $\sum\limits_{n=A}^B$. In that notation, $A$ and $B$ must be natural numbers, and n basically counts from A to B. But in the glyphs above, there is an $\infty$ sign where $B$ was. Since $\infty$ is not a natural number, this is not a valid summation, by the above definition. If we wish to use that notation, we must extend its meaning. And that is exactly what mainstream mathematics has done. We define $\sum\limits_{n=A}^\infty f(n) \equiv \lim_{B\to\infty}\sum\limits_{n=A}^B f(n)$. That definition reached mainstream acceptance because it so closely matches the intuition of most people. So thusly we can say \sum\limits_{n=1}^\infty\frac{1}{2^n} = 1$ because the concept of the limit is fundamentally baked into the use of infinite summation notation. This notation has worldwide acceptance, so your students should learn it.
Likewise, the meaning of the overbar notation for repeated decimals is defined to have a limit in it. If I have the symbol $0.\overline{XYZ_n}$, where $XYZ_n$ is a finite series of n digits, then we define $0.\overline{XYZ_n} \equiv \lim_{m\to \infty}\sum\limits_{p=1}^{m}XYZ_n\frac{1}{10^{np}}$. For our example of $0.\overline 9$, this means $0.\overline 9=\lim_{m\to \infty}\sum\limits_{p=1}^{m}9\frac{1}{10^p}=1$ Again, the concept of the limit was baked into the definition of the symbology. This was done because it helped peoples intuition line up with the precise mathematics, which included a limit.
If you teach them that $0.\overline 9 \neq1$, then you put them in a bit of a bind as students. The greater mathematical community has widely accepted the above definition for what the symbols "$0.\overline 9$" mean. It is almost universal. Instead of challenging that, work with the students to teach them what that symbology means, and show that the limit appears inside the meaning of that symbology.
Just remember, your student's intuition that $\sum\limits_{n=1}^\infty\frac{1}{2^n}$ should equal $1$ is not wrong. It's not bad. That intuition is actually what drove several thousand years of mathematical efforts through arithmetic and set theory to cause us to finally reach the concept of limits. The point of limits is to satisfy that intuition.
