Looking for a nice problem inside SC but not in the first two levels
I don't have a suggestion for a natural problem in $\mathsf{DTimeSpace}(n^{O(1)}, \log^4 n)$, but I do have a suggestion for your side question, of why finding such a problem seems difficult. I think this has something to do with the folk-idea that people can only really comprehend (or maybe are only interested in? or both?) math that is a few quantifier alternations deep. For example, the definition of limit is two quantifiers deep (for all epsilon there exists a delta...); the definition of "$L \in \mathsf{NP}$" is two quantifiers (there exists a machine such that for all inputs...), and the statement "$\mathsf{P} \neq \mathsf{NP}$" is three quantifiers deep.

With regards to $\mathsf{PH}$, this is somewhat borne out by the fact that there are lots of natural problems that are $\mathsf{NP}$-complete, many natural problems that are $\mathsf{\Sigma_2 P}$-complete, and only a few known natural problems that are $\mathsf{\Sigma_3 P}$-complete (see the compendium by Schaefer and Umans). The most natural problems known to be complete for higher levels of $\mathsf{PH}$ come from logic itself, which is less surprising since within a given logic one often has the notion of "$k$-many quantifier alternations," or at least some natural way to simulate it. These perhaps fall into the same category as "accepting problems for NTMs," which you've declared "not nice enough" for this question.

It might also be worth mentioning that the same thing happens in the world of computability, which maybe suggests that it has to do more with our understanding of alternating quantifiers and less with complexity per se. Lots of natural problems are known to be $\mathsf{\Sigma^0_1}$-complete (equivalent to the halting problem), and many natural problems are known to be complete for the second and third levels of the arithmetic hierarchy. But as you go to higher levels of the arithmetic hierarchy fewer and fewer natural problems are known to be complete for those levels. I'm not sure I know of a natural problem complete for $\mathsf{\Sigma^0_4}$, and I've never heard of a natural problem complete for $\mathsf{\Sigma^0_5}$ (though maybe there is).

With regards to polylogarithmic space bounds, I think a similar reasoning applies, but even more so. Since $\mathsf{NL} = \mathsf{coNL} \subseteq \mathsf{DSPACE}(\log^2 n)$, even problems that are in the "first few" levels of the "$\mathsf{NL}$ hierarchy" are all in fact in $\mathsf{NL}$ (the hierarchy collapses), which is contained in log-squared space.