Why is one often requiring space constructibility in Savitch's theorem?
This is elaborated nicely in Dexter Kozen's Theory of Computation textbook, in chapter 2.

Savitch's Theorem (Theorem 1 in his paper) says: if $S(n) \ge \log n$, then $\text{NSPACE}(S(n)) \subseteq \text{DSPACE}(S(n)^2)$.  Space-constructibility often seems to be assumed in a proof, but this requirement can be removed by restarting the search with a fixed space bound that is allowed to increase with each attempt.

The confusion perhaps arises because the original Savitch paper is largely about investigating whether $\text{NSPACE}(S(n)) \not\subseteq \text{DSPACE}(S(n))$.  It therefore spends a lot of effort on space-constructible functions, due to the following observation from the paper:


  It is natural to ask if there is any storage function whose deterministic and nondeterministic complexity classes are equal. The answer was given by Manuel Blum and is "yes". Blum showed that there are arbitrarily large storage functions L(n) such that a set is accepted within deterministic storage L(n) if, and only if, it is accepted 
  within nondeterministic storage L(n). These functions L(n) are not, however, "well-behaved" and Theorem 3 does not apply to them.


(Here "well-behaved" refers to the space-constructible functions, called measurable by Savitch.)