So version 1.0 isn't perfect. Why don't you call the regional health office and make a feature request that ensures families all go in at once.
Also, I suspect you aren't actually familiar with FastPass. Tickets all collected together (such as with families) all have the same return time. Are you sure it isn't the same in Quebec, or are you letting your bias lead you to make assumptions?
Do you have evidence that people are "bored", or is this yet another assumption? And remember, the plural of anecdote is not data.
Funny, my county's health department just set up a number where you call and set up a reservation to come in. Why go through all the trouble of making people drive to the clinic twice?
Why does it have to be some specialized high volume custom phone system? I don't think my local government did any such thing.
So people can't get through instantly and some probably can't ever get in until demand lessens and inventories increase. How, statistically, does that differ from putting a deli-style number dispenser on a corner and saying "go for it!" to a city full of people?
One advantage is that you can serve more people at the same time. It is probably much faster to push a button and get a ticket, than fighting a phone system
It might be faster to push a button and get a ticket, if you're already standing in front of the machine. How are these people all getting to the clinic? Are they driving? Taking public transportation? Cabs? Where are they parking?