Any theory that can represent all recursive functions has no consistent decidable extension, however there are such theories that do not interpret even as weak an arithmetic as Robinson’s theory $R$, see my paper Recursive functions and existentially closed structures (arXiv:1710.09864 [math.LO], to appear in Journal of Mathematical Logic).
You don’t even need to represent all recursive functions. Fix a recursively inseparable pair of r.e. predicates $A,B\subseteq\mathbb N$. Let $L$ be the language consisting of one unary predicate $P$, and constants $\overline n$ for every $n\in\mathbb N$, and let $T$ be the (recursively axiomatizable) theory axiomatized by $P(\overline n)$ for $n\in A$, and $\neg P(\overline n)$ for $n\in B$. Then $T$ has no decidable consistent extension, and it does not interpret much of anything (in particular, if $S$ is a theory in a finite language interpretable in $T$, then $S$ is also interpretable in a finite-language fragment of $T$, and as such does have a decidable extension).