Might it be worth applying to Oxbridge for another subject anyway? (Not sure how different the options are).

I think it is worth investing the time into researching and applying to US Universities and Scholarships. If you can't afford to go, then don't. But the really top schools (Harvard, Princeton, Yale, Stanford, etc.) are often free/heavily discounted for families making less than $100,000 per year. Seems like it makes sense to at least figure out how much it will cost to go. 

My intuition is there is a huge boost in people's job prospects by attending a high reputation school. You'll have better professors, a better professional network once you graduate, and an easier time getting a job in the USA, which seems like a higher impact place to work than may other locations.

Doing an Oxford Math/Phil degree seems good for economics (and I know multiple people who have extremely successfully followed this route). Doing an Oxford PPE degree seems good for politics. Doing Oxford Math or CS seems good for technical AI Safety. I'm less familiar with Cambridge, but assume they have similar degrees available. So wouldn't rule out Oxbridge just because you can't major in Econ. 

All of these opinions are my own and do not reflect those of my employer.