Also for me it stinks like an issue with the SD card (you don’t have to care about the format since while OSMC installation partitioning and formatting is done anyway).
I suggest to
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format the SD card with an appropriate tool on your PC like SD Memory Card Formatter | SD Association
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test (!!!) the SD card on your PC; I personally prefer this tiny program for such jobs H2testw | heise Download (sorry, the web page is in German but the meaning of the green button “download” should be understood)
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reconsider and check whether you want an automatic basic check and repair of the file system on every start of OSMC (don’t know whether this is already worked in to OSMC distribution with Krypton for Pis). See i.e. Automatic fsck of root filesystem on start/stop