Why don't we use larger classes to study determinism vs non-determinism?
The issue may be a bit cleaner with $E=Dtime(2^{O(n)})$ and $NE=Ntime(2^{O(n)})$.
The easiest way to think about these classes is that they are the same as $P$ and $NP$ but restricted to unary languages. That is, all inputs are of the form $1^k$. 

That is, the language $L$ is in $E$ if and only if the language $U_L = \{ 1^x : x\in L \}$ is in $P$ (identifying strings with numbers using binary representation), and similarly $NE$ is isomorphic to unary $NP$.

So, trying to separate $NE$ from $E$ is just like trying not just to separate $P$ from $NP$, but actually do it using a unary language. No reason it should make your life even conceptually easier.