My son's handwriting is terrible and he is in Primary Four. I know things have changed since I was at school, but surely neat writing is still important.
Certainly, the days have gone since students needed to produce reams of beautiful copperplate writing in pristine copy books. And though you do not say when you were at school it is only until fairly recently that neat handwriting was a universal expectation in the classroom.
As you say, things have changed, and it is as common nowadays to see students tapping away at a keyboard as poring over a piece of paper, pencil in hand. And of course the need to use word processors is crucial in the modern world. Many schools even teach typing skills, once the domain of secretarial classes. A lot of exams now allow the use of technology, and using a computer reflects the world of work and leisure outside school.
However, the ability to be able to write legibly is still far from outdated. Taking notes, brainstorming ideas and compiling lists are just some of the tasks still best suited to traditional methods. But of course, there is a difference between legibility and the neatness you mention.
Educators know that the successful manipulation of a writing tool carries many advantages. It encourages hand-eye co-ordination and fine motor skills, for example, and can aid the imagination by not restricting writing to the more linear and restricted template of much of the word processing software.
Handwriting can also aid creativity, allowing students to move quickly from one place on a piece of paper to another and add notes or a rough sketch.