It's probably an aid for attentive repetition. If you can do it faster without writing, great. Many people can't.
(Basically it's a consequence of ADHD. While it's a novelty it's easy to focus, connect it to other things, grasp the rough outline of the concepts, mash them together, be creative, talk about it, barf up ideas about it ... but when it comes to memorizing a proof glyph by glyph reading does not help, because it just makes me feel like reading something that I completely understand and remember the 1000th time. Of course it turns out without actually writing it down from memory it's hard to judge how well I could write it down from memory based on what I feel while reading it...)
There's quite a bit of research linking writing with reading and understanding. Writing to Read: A Meta-Analysis of the Impact of Writing and Writing Instruction on Reading https://doi.org/10.17763/haer.81.4.t2k0m13756113566
I'm aware of this kind of research existing. I guess my personal experience does not count as a scientific work. However it's not study on math and math is not normal reading. It's anything but. You need to understand every symbol, place it in context and do all of the proofs in your head so you can get some feel for it. If you are copying you are not doing this. I'm a terrible hand writer. I have a math degree.
Oh, but everything I know about learning mathematics tells me that doing problems, on paper, is key. That means writing them out. I'm sorry you think you're not a good hand writer, may I recommend Write Now: The Getty‑Dubay Program for Handwriting? You're probably not bad at it, but could use a little refinement. It really helped me improve and speed up my handwriting.
(Basically it's a consequence of ADHD. While it's a novelty it's easy to focus, connect it to other things, grasp the rough outline of the concepts, mash them together, be creative, talk about it, barf up ideas about it ... but when it comes to memorizing a proof glyph by glyph reading does not help, because it just makes me feel like reading something that I completely understand and remember the 1000th time. Of course it turns out without actually writing it down from memory it's hard to judge how well I could write it down from memory based on what I feel while reading it...)