The assumption you are fighting
Almost everyone who has ever seen a hygienist got there the same way. They went to a dentist, the dentist finished, and someone at the desk booked them in with the hygienist down the corridor. That is the only model most patients have.
So when an independent hygiene clinic appears in their search results, the first thought is not "how much" or "when". It is "am I allowed". And because nobody wants to phone a clinic and be told they have come to the wrong place, a share of your visitors simply close the tab.
You never hear from those people. That is what makes this the most expensive problem on a hygiene clinic website and the least visible.
Where the rules actually sit
Whether a patient can come to you directly is a regulatory question, not a marketing one, and the answer depends on where you practise.
In the United Kingdom the General Dental Council has permitted direct access since 2013, so patients can see a hygienist or therapist without seeing a dentist first, within the registrant's scope of practice, and some procedures still sit outside that. In the United States it varies state by state, with different levels of supervision written into each state's dental practice act and rules set by the state dental board. In Canada the position is set by the provincial or territorial regulatory college and differs between provinces. In Australia the Dental Board of Australia sets the scope of practice standard under the national registration scheme.
That variation is the point. Treat everything above as orientation rather than advice, check the current position with your own regulator, and write your website's wording from what they say rather than from what a competitor's site says. Getting this line wrong is a professional problem, not just a conversion problem.
Where the sentence goes
Once you know your position, the sentence belongs in four places.
The first screen, next to the booking button, in the plainest words available. Not in a rotating banner and not below the fold.
The booking page, repeated, because a visitor who arrived from a search result may never have seen your home page.
Its own short page, for the patients who want the detail and for the search term itself, which people do type.
The booking flow, as a confirmation step, so nobody arrives having misunderstood.
Write it for a patient, not for a regulator
The instinct is to write something careful and technical, and careful and technical reads as hedging. A patient scanning on a phone needs the answer in the fewest words your position allows.
Something in the shape of "You can book with us directly. No referral from a dentist needed." Then, underneath, a line acknowledging the thing they are worried about: that you will tell them if anything you see needs a dentist, and how that works.
Say what you do with what you find. That is the second half of the fear and hardly any site addresses it.
Where a referral or an examination is required
If your regulator, your insurer or your own policy requires a referral, a prescription or a recent dental examination, the sentence changes but the placement does not. State it just as prominently, then build the collection into the booking flow rather than into a phone call afterwards.
Ask for it at the point of booking: an upload, a practice name, a date of last examination, whatever you actually need. A visit that has to be rescheduled because the paperwork was missing costs you the slot and costs the patient their afternoon, and it is the sort of thing that produces a bad review about something that was never clinical.
Do not answer clinical questions on the page
Once a site starts talking about who can see whom, the temptation is to explain what a hygienist can and cannot treat, and then to explain what particular symptoms mean. Stop at the first part.
Describe your scope, describe what happens if something needs a dentist, and route anything that reads like a symptom into an appointment. A website is a good place to explain a process and a poor place to give clinical guidance to somebody you have not examined.
What changes when you fix it
Two things, and only one of them is measurable. The measurable one is the share of visitors who reach the booking flow, which usually moves quickly. The unmeasurable one is the phone calls that stop being "do I need a referral" and start being "have you got anything on Thursday".
The visitors who left silently do not come back to tell you. They just stop leaving.
Where this sits
Direct-access messaging on the first screen, the referral path built into the booking flow where one is needed, and a short page giving the detail are standard in a dental hygienist website, and the same care over regulator-driven wording runs through our dental practice sites. What the packages include is on pricing. The sentence itself costs nothing to add. The assumption it removes is the one keeping your diary emptier than it should be.