A fitness studio website has one job: get the schedule in front of the person and let them book from it. Not a link to a booking platform. The schedule itself, on your domain, with the class, the coach, the time, and the spots left, and a book button that finishes on the same screen.
The second job is the intro offer. New members arrive through a first class or a two-week trial, and that offer needs its own page with the price, what it includes, and a checkout that creates the account and books the first class without a phone call. Ads and the Instagram bio link send people to that page, not the homepage.
Then the parts that keep members: instructor pages for the coaches people follow, a member app in the browser for schedule, waitlists, and billing, and texts that remind before class, fill the waitlist, and reach a member who has gone quiet. The scheduling software you already run, Mindbody, Mariana Tek, Momence, Zen Planner, or WellnessLiving, stays. The site sits on top of it.
Underneath: a page for each class type you teach, a page for each city or neighborhood you draw from, LocalBusiness schema, a Google Business Profile that matches the site, and your membership terms stated where members expect them. That is what puts an independent studio in the results for "pilates near me" beside the franchise with the ad budget.