Can I be charged for dental services that were never provided?

Can I be charged for dental services that were never provided?

No. At this point the company's/dentist's delay of service is indicative that they are not planning on fulfilling their obligation toward you[r daughter].

The receipt you signed can be only construed as a confirmation that you authorize the company to charge you. However, that authorization obviously is premised on the delivery of a service which was scheduled for a certain date and which hitherto they have neglected/failed to provide.

The receipt cannot be construed as a confirmation of having received dental treatment. The initial statement outlines three alternatives for the timing of service. Note that the second alternative is the one-word, open ended "will" (should read "will be"), whereas the word "today" pertains only to the third alternative because it is placed between "is/are" and "being", both of which unequivocally pertain to the third alternative. Indeed, "[service] will today being performed" is grammatically incorrect. The term "today" would have to be placed differently (example: "HAS/HAVE BEEN, OR WILL BE, OR IS/ARE BEING Performed TODAY") for a reasonable person to construe the term "today" as plausibly encompassing --or being common to-- all three alternatives.

Accordingly, the delay gives you the option to rescind the contract. Had you already paid, you would have claims of breach of contract, fraud, and unjust enrichment. But paying at this point for a service that has not been provided, and for which they are not really following up, is risky because it might afford to them the allegation that your delayed payment evidences your agreement or acquiescence that the failure to provide the service is your fault rather than theirs.

The amount of $864 reflects that you can file a complaint only in Small Claims court (based on the company's website, I'm assuming your jurisdiction is somewhere in the US), and thus prevent the company from moving forward with debt collectors and/or other actions. Additionally, you might want to report this to the agency in charge of consumer protection in your jurisdiction, since their pattern of conduct constitutes one or multiple unfair and deceptive practices. Most or all jurisdictions have legislation that sanctions businesses' unfair and deceptive practices.

I assume that you signed this without understanding what you were signing, so you weren't trying to deceive the company. The thing you signed ask you to assert that the procedure has already been done, or that it is about to be done that day, neither of which is true. Perhaps you intended to do so that day but something came up. Excluding the case of an emergency, you made a false statement, which is a bad thing. A worst-case scenario theory is that you were involved in a fraudulent arrangement: one of the reasons for you signing this statement is to get you to commit to the position that the work was done or would be done, so that the company can proceede with payment processing.

You now have a contract, and your obligation is to pay up. They have an obligation to pay the dentist, and the dentist has some obligation to perform the job. There is no unilateral option to cancel a contract, so you can't just say "I take it back". You might be able to negotiate with the company, but it looks like you have actually fulfilled your part of the contract (you paid). I am guessing that the company likewise fulfilled their part of the contract. So it looks to me like this is a done deal, everybody has performed, at least w.r.t. you and the company.

So if you want a refund and the dentist is not replying, you may need to get a lawyer. This at least smells like fraud on the part of the dentist. It's probably that your state's attorney general office has an office that investigates such complaints, but it's also likely that any money you get from a complaint will be too little, too late. The main gap in your story is, why didn't you do the procedure?