Oddly enough, I could not find any example of NP-hardness reduction done directly by modeling the problem as a language, and showing that a deterministic Turing Machine cannot decide whether a given instance belongs to that language (I might've messed up with the terminology here)
That's not odd at all: it's because no such proof exists. Anything that can be computed can be computed by a deterministic Turing machine. Any proof that no polynomial-time deterministic Turing machine can decide some problem in $\text{NP}$ would constitute a proof that $\mathrm{P}\neq\mathrm{NP}$, resolving the biggest open problem in computer science. There are proofs that polynomial-time deterministic Turing machines can't decide certain classes of $\text{NP}$-hard languages: see the time hierarchy theorem. However, if $\mathrm{P}\neq\mathrm{NP}$ then, it is a consequence of Ladner's theorem that there are problems that cannot be solved by polynomial-time deterministic Turing machines but which are not $\text{NP}$-hard.
To me, the fact that you considered this as an approach, and that you think there are no problems to reduce from (as distinct from no convenient problems), suggests that you don't actuallyfully understand what $\text{NP}$-hardness is. I suggest you check out our reference question on the subject.