I have a slightly different take on the dentist's office requesting a call-back to confirm you will actually show for your appointment. (Although I don't quite "get" the fee charged structure - seems unusual.)
We have some patients at our clinic who just don't bother to show, even if they've had reminder calls (and often then expect to be "squeezed in" at a time convenient to them, no matter how busy we are).
So, with some repeat offenders, policy has been changed in a couple of ways. Either, we do not book them ahead, but require them to call on the day, and if we can find an appointment for them, then that's fine. If we can't, then unfortunately, that's a natural consequence of past behaviour, where they've not shown repeatedly, which means that people who were on the waiting list could have been seen, but we were unable to do so because someone else didn't show, and didn't call to inform of that.
Or, we will book them in, but upon calling to remind, if a message has to be left, we ask that they call us back to confirm their appointment, otherwise we will not be able to hold it for them. Again, this is with REPEAT offenders, who have an extreme record of not bothering to show because they "forgot", "something came up", "didn't realise the time", "oh, I felt okay, so I didn't worry" and basically didn't care about the inconvenience to us, or to other patients who were desperate to be seen (and there's only so much emergency wriggle room we can manage, even if we don't take a lunch break!)
In some cases, prior to enacting these requirements for certain patients, they have had (at their request!) a phone call the day prior, a phone call the morning of the appointment, followed by a phone call a half hour before the actual appointment time - guess what? They STILL didn't show, and there was no acknowledgement of us going above and beyond to try and help them out. No apologies for waste of time and basically doing a "dog-in-the-manger" of a viable appointment time.) It got to a point where we said "You know what? We are not their mothers." And we're not. We definitely go above and beyond for our patients on a regular basis, but a point comes when it gets ridiculous, and others are missing out because of the selfish actions of a few. (Yes, you can probably tell I'm a little passionate about this - people are in pain, needing help, and we do what we can. However, we need others to play their part too, by using a little common courtesy and consideration.)
So, I can see where the dental office may have had the same pattern with multiple patients, and has enacted this policy to try and best manage their times for people who really need the appointments.
Just my view from the flipside. May or may not resemble the actual scenario faced by the dental office, but often, people simply don't think (or care about) how their actions (or inactions) are affecting others.
On the other hand, life throws a curve-ball sometimes, and if patients have legitimate reasons for not attending, it's never a problem, and we'll do our utmost to accommodate them asap for rescheduling. It's about using resources as best as possible, and being fair to everyone. We would never penalise someone who genuinely forgot as a one-off, or who had other things come up which caused difficulties, but would work with them to help them out. It's just the few, the extreme, that have a disproportionately large impact!