> I’m not sure if words ending in -archy are always pronounced with a hard c but it seems like it.

When it is etymologically derived from the greek arkhein (“to rule”) like all of your examples (but unlike, say, “starchy”, which also ends with -archy but has an unrelated etymology), yes, the “ch” sounds like “k”.

But the “-archy” in “omarchy” doesn't come from arkhein, in comes from Arch Linux, so...