Does he wish to work in the UK as well? I do think it is harder to land internships and jobs in the US from a UK school, even Oxbridge.
Though if he goes the math route, the UK probably would set him up better for grad school as all he’d be doing (if he reads only maths) is math.
He would probably come back to the US.
I think your statement is probably true apples to apples – say, Oxbridge v. HYP. But since any top-ranked US private school is out for us financially, I wonder if that’s still accurate for, say, Oxbridge v. Ohio State. I really don’t have a clue. Or as @PurpleTitan says, if that would still be true for grad school.
Trying to get an offer from an American employer from Oxbridge may be more difficult (definitely more so than from an Ivy). Getting in to an American grad school from Oxbridge definitely would not be (any more than it would from an Ivy). American grad schools take in foreign students from foreign schools all the time. Being an American would only help compared to Internationals at a foreign school.
England could be something to aim for.
BTW, I think the worst he would do state-side is UW-Madison (though I suppose tOSU would give bigger scholarships).
FWIW my daughter just landed a one year sandwich year (year 4 of a 5 year program) position in the UK with a multinational corporation that also has a large presence in the US. We are hopeful that will help with the job search after she graduates as other companies here in the US will be very familiar with her UK employer. So doing internships and such in the UK, which is allowed on student visa, is one way to mitigate the effect of a foreign degree.