Powers under the Constitution are generally read to be complete within a particular ___domain except to the extent they are counter-balanced by affirmative rights. See Gibbons v. Ogden ("If, as has always been understood, the sovereignty of Congress, though limited to specified objects, is plenary as to those objects...").

Yes, but you missed the part that "common Defense" occurs in the preamble discussing purpose, not powers, and in a phrase limiting the purposes of the Taxing power. Its not itself an enumerated Congressional power (Congress does have more narrow enumerated defense related powers), so its irrelevant, in discussing it, that grants of power to Congress are generally plenary except within express limitations.

Look at another clause in that section:

"To exercise exclusive Legislation in all Cases whatsoever, over such District (not exceeding ten Miles square) as may, by Cession of particular States, and the Acceptance of Congress, become the Seat of the Government of the United States..."

By your method of interpretation, this clause gives Congress the right to exercise legislative jurisdiction over D.C., but not the right to build the city. A more sensible interpretation is that it grants Congress the power to build the city in the same breath as granting it exclusive legislative jurisdiction over it. The embedding of the power to provide for the common being embedded within the power to tax can be seen as operating similarly. Note that this drafting reflects history. There was no doubt under the articles of confederation that Congress had the power to provide for the common defense. The sticking point was that it did not have the power to tax to achieve that end.

Also, the preamble is not legally inoperative. It can be read as establishing "common defense" as an umbrella term for powers mentioned within.

Finally, providing for the common defense was considered at the time and today an inherent aspect of sovereignty, and the Constitution through the preamble shows the intent to invest that aspect in the federal government.

I can see the point you're getting at, but I don't think it works. You're comparing a power that's mentioned twice in the Constitution (archaic drafting aside), was considered an inherent function of government at the time, and was clearly motivated by by contemporary events as supported by the historical record, to an umbrella "right" that isn't mentioned explicitly by the Constitution, isn't a neat generalization of the rights that are explicit in the Constitution, and wasn't part of the historical understanding at the time.

I think there's some Occam's Razor analysis that needs to happen here. The simplest explanation of the text is that the common defense was intended to be a fundamental function of the federal government, while the various rights that arguably go to privacy were really derived either from a response to specific injustices at the hands of the British or from common law understandings of due process rather than some unstated general umbrella right of privacy.

> By your method of interpretation, this clause gives Congress the right to exercise legislative jurisdiction over D.C., but not the right to build the city.

No, it doesn't. Legislative jurisdiction over the territory itself includes the ability to build the city.

> You're comparing a power that's mentioned twice in the Constitution

The phrase is mentioned twice, but neither is a grant of power (one is a statement of the overall purpose of the Constitution, the other is expressly a limitation on the scope of another grant of power.)

There is no mention of "common defense" as a power anywhere in the Constitution.

> No, it doesn't. Legislative jurisdiction over the territory itself includes the ability to build the city.

No more or less than the power to tax for the purpose of providing for the common defense includes the power to actually provide for the common defense.

Which it doesn't, any more than the power to tax for the purpose of general welfare (which is part of the same phrase as the "common defense") gives Congress independent non-taxing power to provide for the general welfare outside of the grants elsewhere in the Constitution. There's quite a lot of conditions, etc., you can apply to liability for taxes, etc., that allow some substantive regulation to be plausibly included under the taxing power, but if you read the purpose limitation of the taxing power to instead be a positive grant of power independent of the taxing power, as you suggest, that one clause alone would shift the federal government under the Constitution into one of universal plenary power with only negative restriction instead of one of specific positive powers, and its quite clear that that was never the intent of that clause (as well as that interpretation being completely inconsistent with the actual words of the clause.)