"No appointments available" for request through online booking is extremely explicit.
I find that when potential/existing clients see this they believe either that there are absolutely no appts available that day or believe that the requested staff is fully booked on a day they are not even scheduled.
It would be better to let them know that there are other service providers available on that date to attempt to retain the potential booking and not be worded so absolute that there is no hope to get an appointment on the selected date.
Additionally the flow implicitly causes the client to assume that the service provider was or is fully booked on a day they are not scheduled. At date selection earlier in the flow, perhaps dates that are not scheduled for that provider would be greyed out and non-selectable to denote that this provider is not working those dates. This would also in turn solve the issue of client side booking waitlisted appts on dates that are not scheduled for those providers.
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Carolyn Nizenkoff commented
When you have calendar set to only have 30, 60, 90 days available for online booking when I client goes to look for a date beyond that instead of saying nothing is available have it say the date isn't open for booking yet.
similarly, when they are looking at a day that no one is scheduled to work, instead of saying no appointments available having it say business closed or something similar.