Our wiki mentions (in early footnotes to US admin_level) that the Census Bureau “sorta” does this as it gloms DC into a county equivalent. It isn’t (in reality), but it is “convenient” that under certain circumstances our Commerce Department does this. Our wiki explicitly say OSM chooses not to. That’s at least something (that clarifies, and says “bureau does this, OSM does not.”)
So, “Washington, D.C.” is, as our wiki (in United States / Boundaries) says admin_level=8 and admin_level-4, respectively. This is in a section called (I believe Minh-titled) “Notable Exceptions.” I’ve been OK with this, I’m glad to see us talking about it and heads nodding (or continuing to nod lightly). Fully nodding? Unanimously nodding? Well, we’re getting there.
So to sum up, no, that logic says that we don’t “promote” the nation’s capital to admin_level=6. The bureau might (and does, for certain purposes), OSM doesn’t. And we say so. And heads have been (at least lightly) nodding for some time about that.
Does DC “go straight from 4 to 8?” Yup, it does. It’s odd, but it’s explainable; we do explain.