There's clearly a line there somewhere. Or at least, there's one to me. On the one hand, we've been playing around with automatic assessment for ages in both testing and instruction. Intelligent Tutors, Cognitively Based Instruction, Mastery Learning, all of that is based around the idea that the computer can provide some guidance to the learner as they navigate the space. There's plenty of room for automatic something-or-other.
Let's look at essay scoring as a special case of the Problem (capital-P problem): a complex student response to a particular challenge. In some domains, this is a serious (actual) problem: proving a theorem, analyzing data, designing an experiment, designing a user interaction, creating and supporting an argument.
In the case of the Problem, I do not like any kind of real automatic scoring. To be flippant, I'd say that anything that is amenable to auto-coding isn't really a Problem, it's probably an exercise (writing a five-paragraph essay on the plot of a book is an exercise, arguing about the motivation of a character in the historical context of the novel is a Problem).
To be more thorough, I think that any student who works on a real Problem needs frequent and supportive formative assessment. Automatic coding might be fine for discriminating Good from Bad, but I doubt it would be able to provide substantive feedback to the student. To that end, in the inquiry-support software I did my dissertation on (and am currently re-writing from scratch), I ask students to assess themselves and each other, instead of trying to assess the quality of the hypothesis automatically. That's a digression, though, so I'll get back to the main idea.
I could imagine an auto-scorer that let you frequently assign Problems, but only manually grade (and provide feedback on) the weakest students' work as well as some rotating randomly-selected other student work so you can make sure you're giving the most help to the people who need it most.
I would still rather the answer be: adjust teacher workloads so that they have a small enough total student size and sane enough content workload that this is a non-issue. If you can have N students work through a lot of interesting Problems as they learn Foo and provide feedback to them all, that's a good class size and set of standards. If we have to cut corners on feedback to either cover the material or triage the especially needy, that's something that needs improvement.