New belt is tighter/grippier and stouter/stiffer than the old belt. It's also more angular in the teeth verus a worn belt (they're more rounded). It's also shorter. It will put new pressure against the old components which they haven't had to deal with before in the whole chain of timing and cooling. If you change the cooling gearing surface and the tensioner roller surface you've got new traction points/surfaces versus old traction at older points in the chain. This direct dissonant bias is now tranmitted to the the cam shafts drive, and their gearing and bearing surfaces, that by now have most certainly degraded. It's disharmonius at best, it' s the new choir trying to sing with the barbershop quartet of old sort of thing. This guy I work with is an ace Ferrari/ Lambo/ Porsche mechanic he wouldn't steer his customers wrong.
So using that logic, if the water pump or timing belt fails, you need new cams?
Or maybe even new heads?