There are many metrics $d: \Bbb N \times \Bbb N \rightarrow \Bbb R^{\ge 0}$ with respect to which the natural numbers are a compact space, see footnote$^1$. But there is a possible misconception here that I want to point out, thereby maybe answering your question.
Your idea seems to be something like this: If we can show that "the natural numbers are compact", then every Cauchy-sequence in it converges to some natural number; therefore, if we can show that, for any start value $n$, the sequence $C(n), C^2(n), C^3(n), ..., C^k(n)$ of iterated applications of the Collatz function is Cauchy, then it converges to some number, therefore at least "never goes to $\infty$". Then you start thinking about how to refine this to exclude finite cycles etc.
This is flawed. Why? You write
if a sequence [is] a Cauchy-sequence it always converge[s] if it is located in a compact space.
This is imprecise and the imprecision is the cause of the error. What is true is:
If a sequence is a Cauchy-sequence with respect to a certain metric $d$, and the space in which the terms of the sequence lie is compact with respect to that metric $d$, then the sequence converges -- with respect to the metric $d$ -- to an element of that space.
In short, terms like "Cauchy", "converge", "compact" (as well as: open, closed, bounded, connected, ...) depend on the metric you choose, and something which has one of these properties with respect to one metric $d_1$ has, in general, no reason to also have this property with respect to a different metric $d_2$.
So, when the natural numbers are compact w.r.t. to some metric $d_1$, then you get without work that e.g. every sequence in $\Bbb N$ has a convergent subsequence with respect to that metric. Even the sequence $1,2,3,4,5,6,...$ -- in the metric $d_{17}$ of my footnote, this sequence converges to $17$. Does that show that it "does not go to $\infty$"? No, because by that you certainly mean "go to $\infty$ w.r.t. the usual metric $d_{Euclid}$", and the behaviour of a sequence with respect to $d_{17}$ says next to nothing about its behaviour with respect to the usual order and metric of $\Bbb N$, which is what you are actually interested in.
Let me show this in one example which is similar to what I assumed above as your intended reasoning. Say I want to prove that the twin prime conjecture is false, i.e. there are only finitely many prime twins. For this, I define a sequence $$T_n := \:\text{number of prime twins }\le n,$$ so that $T_1 =...=T_4=0$, $T_5=T_6=1$, $T_7=...=T_{12}=2$ etc. Now obviously the twin prime conjecture is equivalent to the sequence $T_n$ going to $\infty$. But with respect to the metric $d_{17}$ defined in the footnote, w.r.t. which $\Bbb N$ is compact, it is easily seen that $T_n$ is a Cauchy sequence, hence must converge to some number $N$, "so it cannot go to $\infty$", and voila, I have disproved the conjecture. -- Of course I have not. The flaw is this: When you say something "goes to $\infty$", you are tacitly assuming "with respect to the usual metric $d_{Euclid}$". However, the metric $d_{17}$ is made so that something which monotonously goes to $\infty$ with respect to $d_{Euclid}$ will just converge to $17$ with respect to $d_{17}$. So if $T_n$ does not stabilise, w.r.t. the metric $d_{17}$ it will converge to $17$, but this does not mean that there are only 17 prime twins: it still goes off to $\infty$ w.r.t. the usual metric and order of $\Bbb N$. (And philosophically, this should be clear: Because nowhere have I even used anything about twin primes.)
To phrase the possible fallacy in yet other terms: Viewed as a set, the natural numbers $\Bbb N$ have no (geo)metric/topological/order structure at all. It's just a countable set, which is in bijection with any other countable set. We cannot even really imagine that (at least I can't), because if one tries to imagine countably many points, one always puts them in some order (like dots on a line, grid, in a cloud, whatever), and thereby tacitly puts some "nearness" structure on them (some of the dots are close to others, some are far, somewehere maybe some accumulate, or some go off infinitely in one, two, many directions ...). Now the problem is that on the one hand, you want to prove statements about $\Bbb N$ with the usual ordering and $d_{Euclid}$ (dots on a ray starting somewhere and going to $\infty$), but on the other hand, you want to use some other metric -- which means you reshuffle and re-order those dots; in my example $d_{17}$, you now have them all within a finite interval, accumulating towards one of the endpoints, and including that endpoint; or, here is how one begins to order $\Bbb N$ according to some $p$-adic metrics (w.r.t. which $\Bbb N$ is bounded, although not compact).
That all being said, the really interesting question would be: In what cases does something you can prove about one metric carry some information about the same set with a different metric? It is quite possible that there might be some metric/topology/other structure on $\Bbb N$ that would help in a proof of any given number-theoretic conjecture (Collatz, twin prime, Goldbach, Riemann, you name it). However, none of that could be a deus ex machina as in the flawed argument above; that metric would need to involve, in its definition, some deep number theoretic information which maybe surprisingly translates into something we can sort of visualise geometrically. For example, there is the infamous "topological" proof of the infinitude of primes. Looking at the comments there, you will see though that many mathematicians argue that the proof idea can actually be written without topology, and the topological insight is small, or that proof just hides the technicalities well. This is something to be aware of: You cannot avoid work. You can just hope to translate it to something which has already been done or for which tools have been developed. My suspicion is:
If one could show a) there is an interesting metric $d_{dream}$ w.r.t. which Collatz sequences are Cauchy, AND b) convergence in $d_{dream}$ has some implications for behaviour in $d_{Euclid}$ or the usual order of $\Bbb N$, then the hard part would be to show a) and b), and to show that Collatz sequences are Cauchy. Probably, with all that work, whatever proof would show that the sequences are Cauchy in $d_{dream}$ could most probably be refined to showing that they converge in the usual metric, and the whole excursion to metric spaces and compactness arguments would have been useless.
$^1$ Here is a metric with respect to which $\Bbb N$ is compact. I call it $d_{17}$ and use the number $17$ because today it's my favourite number; obviously, I could have chosen any other.
For $m,n \in \Bbb N$, let $d_{17}(m,n) =\begin{cases} |\frac{1}{m}-\frac{1}{n}| \text{ if both } m,n\neq 17\\
\frac{1}{m} \text{ if } n=17, m \neq 17\\
\frac{1}{n} \text{ if } m=17, n \neq 17\\
0 \text{ if } m=n=17.\\
\end{cases}$
Instead of checking that this is a metric and it makes the space compact by hand, remark that all I did was identifying the set $\Bbb N$ with $\{ 0, 1, \frac{1}{2}, \frac{1}{3}, ..., \frac{1}{16}, \frac{1}{18},\frac{1}{19}, ...\}$ (sending $17$ to $0$ and everything else to its reciprocal) which is compact with respect to the usual metric, and pulling back that metric via the identification.
Remark that starting from this, one can build fancier examples of countable compact sets which, exactly by being countable, can all be viewed as $\Bbb N$ with some fancy metric.
