"Constitution OF the United States" That is the corporate one.

The original is "Constitution FOR the United States"

I think you are wrong, they are indeed swearing to the corp (unknowingly).

Edit:

We the People of the United States, in Order to form a more perfect Union, establish Justice, insure domestic Tranquility, provide for the common defence, promote the general Welfare, and secure the Blessings of Liberty to ourselves and our Posterity, do ordain and establish this Constitution for the United States of America.

https://constitutioncenter.org/media/files/constitution.pdf

TITLE & PREAMBLE: TITLE – The Constitution is “for”, not “of”; the word “of” expresses “possession”; in this case, a possession of the United States; therefore, its creation, to do with as it pleases; it denotes ownership. Whereas the word “for” denotes something “intended for” someone else; in this case, a contract of the People, intended for the United States.

https://www.nationallibertyalliance.org/sites/default/files/chapter2.pdf

Also this: http://www.paulstramer.net/2020/04/the-constitution-for-united-states-of.html