It might be pedagogically novel, but I'm confident lots of people learn this way, at least for alphabet-based languages.
From my understanding, Montessori is more about doing + engagement than passive learning, and the emphasis on writing would be a way to express those values.
Most of this stuff is parent directed because they want little Johnny to be brilliant, but afaik goes against pretty much all early childhood development research.
So, no. This doesn't go against any research. Unless you frame it in the most uncharitable way that you can.
Plus given there are three years worth of children in the same class, seeing the “senior” children operate at the higher levels of activities helps with incentive for the younger children who want to emulate them.
That said, anyone can open a “Montessori school” and not adhere to any particular Montessori pedagogy. I’d argue the schools with the purest interpretations of Maria Montessori’s teachings are often just as problematic as those who are Montessori in name only. There are two tribes in Montessori: AMS (purists)and AMI (more flexible).
Thankfully the school my daughter attended was AMI and I watched the school adapt their methods to children who needed more structure than classical Montessori advises. Ultimately, it’s not for everyone.
I don't think it really matters if you start with writing or reading first, because it's basically the same skill: knowing letter shapes and knowing which sound corresponds. It's that learning really benefits from receiving the same information in different contexts. So as long as you both teach them to write and read, the order doesn't really matter.
One reason for using the cardboard letters for younger children is that the brain regions handling bodily movement are much more advanced, and thus provide more context which (as stated earlier) helps in memorization.
But I suspect the main reason for using the sandpaper letters is because it is way more interesting for the children to have something with tactile feedback, instead of just looking at a flat card.
I have no idea whether this method is better, but as a parent it certainly seemed like a very novel approach. Seeing my three-year-old daughter learning to write was (like many Montessori things) surprising.