If "Allow clients to schedule this Appointment Type, but do not create a ticket." is grayed out when setting up the Shareable URL section within an Appointment Type, this is due to the Appointment Type being set up for use within tickets.

To change the functionality to use outside of tickets go to: PSA Options > Select to use Activities/ To-do's instead.

Within the Shareable URL section, you will now see the option to "Allow clients to schedule this Appointment Type via Shareable URL, but do not create a ticket" as an available option.