Well, since you mentioned that I do not need to focus specifically on grammar method, I won't; I will instead focus in a general spectrum based on a previous answer of mine.
This Guardian article explains this very well. The following is quoted from the article (statements made by "Edouard Gentaz, professor of developmental psychology at the University of Geneva").
Some neuroscientists are not so sure. They think that giving up handwriting will affect how future generations learn to read. “Drawing each letter by hand substantially improves subsequent recognition,” Gentaz explains.
The above is an expert opinion of course. But, the article also mentions a study done with 76 people, all aged 3 to 5. The results show that the children who wrote out the letters by hand were better at recognizing them than those who used a computer. The same happened when this was tested on adults:
Marieke Longchamp and Jean-Luc Velay, two researchers at the cognitive neuroscience laboratory at Aix-Marseille University, have carried out a study of 76 children, aged three to five. The group that learned to write letters by hand were better at recognising them than the group that learned to type them on a computer. They repeated the experiment on adults, teaching them Bengali or Tamil characters. The results were much the same as with the children.
The article also states:
Drawing each letter by hand improves our grasp of the alphabet because we really have a “body memory”, Gentaz adds. “Some people have difficulty reading again after a stroke. To help them remember the alphabet again, we ask them to trace the letters with their finger. Often it works, the gesture restoring the memory.”
This thus shows how hand writing is also helpful in attempting to recover lost information, including something as simple as the alphabet. The article again uses a study performed by Princeton University and the University of California, Los Angeles, including 300 students total. It shows that students who wrote out notes on paper did better answering questions during a lecture than those who took notes on a computer/laptop:
The study focused on more than 300 students at Princeton and the University of California, Los Angeles. It suggested that students who took longhand notes were better able to answer questions on the lecture than those using a laptop. For the scientists, the reason is clear: those working on paper rephrased information as they took notes, which required them to carry out a preliminary process of summarising and comprehension; in contrast, those working on a keyboard tended to take a lot of notes, sometimes even making a literal transcript, but avoided what is known as “desirable difficulty”.
I'll also add this ScienceDaily which touches on the same subject. Quoting the quotes I quoted from my linked answer (the following basically came from the above article):
Writing by hand strengthens the learning process. When typing on a keyboard, this process may be impaired. Neurophysiologists have examined research which goes a long way in confirming the significance of these differences. When writing by hand, our brain receives feedback from our motor actions, together with the sensation of touching a pencil and paper. These kinds of feedback is significantly different from those we receive when touching and typing on a keyboard.
and
When writing by hand, the movements involved leave a motor memory in the sensorimotor part of the brain, which helps us recognise letters. This implies a connection between reading and writing, and suggests that the sensorimotor system plays a role in the process of visual recognition during reading, Mangen explains.
