There's data here for number of doctorates awarded by school in 2020: https://ncses.nsf.gov/pubs/nsf22300/data-tables
#1 was... Walden University. Which I'd never actually heard of before. It's a private for-profit online school.
The rest of the top 10 is 2. Michigan 3. Illinois 4. Berkeley 5. Purdue 6. Texas A&M 7. Stanford 8. Texas 9. Wisconsin 10. Ohio State.
So there's not a direct correlation between "eliteness" and volume of doctorates produced. Some of the elite schools are represented in this list, but "non-elite" schools are too.