I'm referencing that same National Academies Press book again, this time the chapter about titanium ores:
https://nap.nationalacademies.org/read/1712/chapter/5
Titanium ore reserves are abundant and widely distributed throughout the world (Figures 7 and 8). U.S. self-sufficiency has declined from 75 percent in the mid-1960s to 45 percent in 1980 because of the ready availability of low-cost concentrates from other countries rather than from a lack of U.S. reserves. In terms of contained titanium, the manufacture of titanium dioxide (TiO2) in the United States is about 15 times greater than the production of titanium metal. An emergency need for additional raw material for metal manufacturing could be met readily by diversion of ore, or of titanium tetrachloride, from the TiO2 industry. Ore cost, about $0.35 per pound of contained titanium, is a minor part of the price of titanium metal.
Titanium ore is not rare, expensive, or difficult to find domestically. Producers of titanium minerals listed later in the report include the United States, Canada, Finland, Norway, South Africa, India, the Soviet Union, Malaysia, and Sri Lanka. As of 1983 Canadian and Norwegian production were higher than US production. Soviet production was lower than Norway, Canada, or the US.
This report was published 20 years after the SR-71 started flying, but titanium minerals were never rare and they have long been produced in the United States and in closely allied countries. That's why I wonder if there is something to explain "The US worked through Third World countries and fake companies and finally was able to ship the ore to the US" or if it's just a story.
To me it reads a bit like a story about how the US tricked the USSR into shipping lead ores to the American small arms industry during the Cold War by saying that deer season was coming up [1]. It sounds clever and funny. But it doesn't make a lot of inherent sense that the US sector would need to clandestinely obtain a common mineral resource from its rival. The very fact that it is good as a story makes me want to understand more before accepting it, because good stories have a way of propagating themselves regardless of their relationship with the truth.
[1] Note: I have never heard this story about lead ores. I just made it up on the spot as an example of a story that sounds so good that I would be inherently disinclined to believe it without further evidence.