Can NFAs determinization be efficient given that the statespace is exponential?
Let me mention a variant of the powerset construction. In this variant, we are going to construct only states reachable from the initial state. The way it works is as follows:


Create a list of states to-be-processed, and populate it initially with $\{q_0\}$, where $q_0$ is the initial state of the NFA.
While the to-be-processed list is non-empty, remove a state $S$, and compute all states reachable from $S$ by reading one character. Put any of these which hasn't been encountered yet in the to-be-processed list.


This variant (which I haven't quite specified in full) constructs the powerset automaton restricted to states reachable from the (new) initial state, and runs in time proportional to the resulting number of states (and the alphabet size).

The same construction, and indeed the vanilla powerset construction, also works for NFAs with several initial states. The only difference is that we start with the set $Q_0$ of initial states (which is also going to be the initial state of the resulting DFA) rather than with $\{q_0\}$.