Not one of the implementations pulled in in BabelMark translates the ~~word~~ syntax into an HTML <s> element (the <del> element type is an obvious alternative), and the majority of implementations don’t seem to recognize the syntax as anything special at all.
And by the way: How do you explain that ~~ should translate into <s> and not into <del> (and one would also need, for consistency, a syntax that translates into <ins> in that case)?
So in my point of view, I doubt that extending the CommonMark syntax in this direction is consistent with CommonMark’s stated goal of being “highly compatible” (although extending and not only refining the syntax has been done before, too), at least for now.
I for one would wish that QUOTATION MARK translates (under certain circumstances) into <q>, which is after all the “official” HTML element type to contain, well: inline quotations—but writing the tags literally into the text does achieve what I want anyway (with just one keystroke more), so I don’t see any urgency here. And the same applies for strike-through formatting, in my opinion.