Why not talk about utility functions on the surreal line when preferences are lexicographic, etc?
The first thing that comes to mind is that some economic models have more than one agent.

If one agent has a utility function with values in ${\mathbb R}$, then two agents have a pair of utility functions with joint values in ${\mathbb R}^2$.  If one agent has a utility function with values in $S^*$, then two agents have a pair of utility functions with joint values in ...what?  We can't just form a product of $S^*$ with itself, because $S^*$ is not a set.

There are probably ways around these difficulties, but they're going to be clumsy and they're going to require lots of mucking around with foundations that are almost surely not interesting to most economists.

Set theory lets us form unions, products, coproducts and power sets.  It lets us talk about subsets and supersets without having to worry about whether they exist.  Economists use these constructions all the time without stopping to think about them.  I doubt that we want to switch to a new foundation that forces us to think about them constantly.