Back in the '60s I was a 'pot boy' in an Italian restaurant. The cooks had a big pastiolla (pot with a sieve-like insert that sits all the way down into the pot) filled with boiling salted* water. When an order came in, they'd dump pasta into the pastiola, put a ladle of hot sauce in a sauce pan, start it bubbling. Then the sieve-thing got lifted out, pasta dumped into the bubbling sauce, cooked for a bit - and the whole business got plated. I never saw them take pasta water and put it into the sauce. Maybe they were relying on the water stuck to the pasta and the sieve? Over the course of an evening the pastiola water would get coudier and cloudier (that sieve was a major PITA to get clean - even after soaking).
*I once drank some pastiola water (on a bet with one of the cooks). It was salty, but not *salty* (soup-salty, not ocean-salty).
FWIW, I was 14. It was a Saturday night job. I don't remember lying about my age, but at 14 I was 6'1" and they probably never asked me. Maybe 14 was OK for a non-school-night job? Dunno.