There is one subtle thing going on that is easy to miss.
What it means (if your treaty is typical) means that you must report the income, but you may get a foreign tax credit to offset the US tax on the foreign income -- but only the US tax and only in the proportion of the foreign income / worldwide income.
You point you the treaty language stating that only the foreign country can take your foreign social security. Great. But often there is a so called "savings clause" hidden elsewhere in the treaty that says the US can also tax everything because of citizenship (not residency).
Often the US agrees to give the a tax credit for such taxation based solely on citizenship. That is the FTC.
So what the treaty provisions are really doing is overriding the US tax code foreign/domestic source rules. Those rules determine what income is treated as coming "from sources with the US." The FTC only applies to income "from without the US" so getting the foreign income to be foreign source is essential to getting the FTC. Sometimes the treaty rules agree with the US code rules, sometimes not. Disclosure is required if they override the code rules and an exception does not apply (you properly cited the complex exception regulations).
For the US source rules see IRC 861 et. seq. and the regulations. https://www.law.cornell.edu/uscode/text/26/861
For an example of the saving clause see the US Spain tax treaty. https://www.irs.gov/pub/irs-trty/spain.pdf (emphasis added)
Art 1(3): 3. Notwithstanding any provision of the Convention except paragraph 4, a Contracting State may
tax its residents (as determined under Article 4 (Residence)), and by reason of citizenship may tax its
citizens, as if the Convention had not come into effect.
Art 24(2): In accordance with the provisions and subject to the limitations of the law of the United States (as
it may be amended from time to time without changing the general principle thereof), the United States
shall allow to a resident or citizen of the United States as a credit against the United States tax on
income
(a) the income tax paid to Spain by or on behalf of such citizen or resident; a
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