Doesn’t directly answer the question, but this was a fairly interesting read about the neutral service at the giro: http://road.cc/content/feature/83410-inside-giro-ditalia-vittoria-servizio-corse-pelotons-guardian-angels
Sounds like the neutral bikes are only ever used very occasionally and very briefly, until they can get a spare team bike, but as others have said if a rider has mechanical when the team car can’t get to them (eg in a breakaway on narrow mountain roads), then the quick change to a neutral bike hopefully avoids the race being decided by mechanicals not riders.
Also interesting that as well as pedals, they record which teams are on which groupset to make sure rear wheels are compatible, so they can get the right wheel without needing to check with the rider first – saves a few seconds I suppose.