The Scribe surface should act like paper. A screen protector is unnecessary. And yes, it can cause this problems.
For many many years I use readers from Sony, Kobo and Amazon. Never a reader had problems with the unprotected screen. They all had or have a case/sleeve for transportation, that's enough for their safety.
Additionally I have a reMarkable 2, also with an unprotected screen (but with a Folio sleeve). I use two different pens, an original reMarkable pen and another from Lamy. The screen looks clean as at the beginning.
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