That's not true. I worked for a big tech company with a large European presence and they 100% heavily factored in the prestige of what university you went into for hiring. Also, they do this with career fairs/college hiring - they aren't sending college recruiters to a community college, they're sending them to MIT, Oxford, Stanford, Yale, Harvey Mudd, etc.
If you work for an American company, you usually get the American experience. Local companies tend to be more in sync with the local culture.
One thing I remember from Finland is people who went to a prestigious foreign university and then returned home without work experience complaining how difficult it is to find a job. In some fields, a fresh graduate without real professional experience beyond internships is already a bit suspicious, because the Finnish system encourages starting your career before graduating. Finding job opportunities is difficult, because you don't have any professional contacts. And while recruiters recognize that you went to a reputable university, they don't know what to expect from you.
And if you went to a school like Harvey Mudd, which isn't really known outside the US, your situation is even worse. Because they don't know the school, recruiters may assume that you got your degree from a for-profit diploma mill and ignore your application.
I agree that things will be more localized depending on the company. But good recruiters should know what schools are good, but they should also be able to assess a resume that doesn't have an elite school attached to it. Obviously this doesn't always happen, but it should in practice.