Defining the halting problem for non-deterministic automata
There are a few reasons I think we put less effort into  the Halting problem for non-deterministic models.

The first is that there are, in fact, two relevant halting problems for a ND model. Given an input $x$ and a non-deterministic machine $M$:


Does there exist a valid run of $M$ on $x$ which halts?
Does there exist a valid run of $M$ on $x$ which doesn't halt? i.e. do all valid runs halt?


For deterministic machines, these are identical, since there is exactly one valid run of $M$ on an input $x$. But for non-deterministic machines, there might be multiple runs. Which one you're interested in depends on your application.

Secondly, non-deterministic models are already unrealistic: they assume that you either have a magical box telling you which path to take, or that you have some form of infinite parallelism. Since non-deterministic and deterministic Turing Machines are equivalent in power, in most cases you just convert the machine into a determinstic one before you worry about halting.

As an extension of this, we don't care because proving something about a non-deterministic machine is at least as hard as proving something about an equivalent deterministic machine. We already know that there's no solution to the deterministic Halting Problem, so all it's really useful for is proving other problems undecidable through reductions. And it's always going to be less work to reduce to the deterministic halting problem, since it's easier than it's non-deterministic counterpart.