Help me resolve my doubts about infinities, exemplified by an example of non DFA acceptable language
Look at your general procedure: given a language $L$ and integer $n$ build a DFA that properly recognizes any string $x \in L$ with $|x| \leq n$. Note that the language you selected is not special, any language can be represented by a sequences of DFAs with one for each $n$. To see this, just build a tree DFA with $|\Sigma|^{n + 1}$ states with a state for every possible string of length $\leq n$ marking the ones in $L$ as accept states and the ones not in $L$ as reject states and then minimizing. (If you want an even simpler example, then note than for any $L = \cup_{x \in L} \{x\}$ and every $\{x\}$ is recognized by a DFA with $|x| + 1$ states)

The whole point of DFAs is that it is a finite object that recognizes an infinite object. In fact, this is the whole point of computer science in general: Turing Machines are a different kind of finite object that recognizes (or encode) a different (larger) set of infinite objects. This means that for a language to be recognized, you need a single machine that works for all strings in that language.

What you have shown is that $L_n = L \cap \Sigma^{ \leq n}$ is recognized by a DFA, but these are just finite languages and every finite language is recognized by some DFA. However, regular languages are not closed under infinite union (in fact, very few things of interest to computer scientists are closed under arbitrary infinite unions), so the simple fact that $L = \cup_{n = 1}^{\infty} L_n$ does not imply anything about $L$.