Is $NSPACE(S(n)) \subseteq DSPACE(S(n))$ if $S(n)$ is time-constructible?
Space-constructibility is a red herring. The real problem in your argument is the step
$$
\mathsf{DTIME}(c^{S(n)}) \subseteq \mathsf{NTIME}(S(n)),
$$
which is not expected to hold in general. For example, the problem of deciding whether an input Turing machine halts in time $2^n$ can be solved in deterministic exponential time, but is conjectured not to be solvable in nondeterministic polynomial time.

Furthermore, while it's not clear whether Savitch's theorem is tight, it is expected that in general $\mathsf{DSPACE}(s(n)) \neq \mathsf{NSPACE}(s(n))$. For example, it is believed that $\mathsf{L} \neq \mathsf{NL}$.