Nah it should definitely stick to handwriting. When you handwrite, you're limited in the sense that you can't waste time going over and editing etc, by rearranging phrases and all. Plus it would be a bit too easy to memorise an essay, type it out, and change the topic sentences to adapt to the question– doesn't test your ability to think on the spot with time pressure as well as handwriting does.
Right now people argue that handwriting is unfair because some people write faster than others, but I don't think computers will change anything because very soon there will be complaints about how there are so many variations in typing speed. And whilst handwriting speed is something that anyone should (theoretically) be able to improve on, typing would pose a disadvantage to people who aren't familiar with computers (e.g. rural and remote locations etc).
Disadvantages aside, I think we're getting too technology-based (and not trying to sound like I'm 80), but seriously–
we've already replaced mental maths with calculators, research books with the internet... and if we replace our hands with computers, then there aren't many more steps left before we'll be replacing our brains with something else too. I think English texts in particular reveal a lot about what happens when humans mess around too much with technology, think of Frankenstein/Blade Runner for example