Shopify was not setup for schedulable services, so for not it's all workarounds or apps.
Bookthatapp is one scheduling app, zapiet is another app more focused on delivery scheduling.
You could manage the times with shopify-flow if you can add metafields to the shop[1] or products, or even a collection object that have a corresponding update metafield action.
Then display the times from the metafield using liquid to put them either in line item properties customization or the cart attributes customization to carry through in the order data.
Then when triggered by order-created action or the inventory-quantity-changed action use the update-shop-metafield action to disable the available times by checking the custom attributes on the order.
Alternatively you can cobble this using similar technique to what your already doing with variant inventory in combo with different advanced customizations.
I'd first try and move the time windows to it's own free product, and it's on that that customers must select the time.
That way you can reset the inventory every day to 1, using CSV, the bulk editor , or try and automate with shopify-flow (example scheduling product changes) .
Then present the appointment product/variant in the cart page.
Modify the gift-wrap tutorial to present a drop down of time options
https://shopify.dev/tutorials/customize-theme-add-gift-wrap-option
And/or combine it with a date picker https://help.shopify.com/en/manual/online-store/themes/themes-by-shopify/vintage-themes/customizing-...
And make it mandatory that a time(a variant) is selected
https://community.shopify.com/c/shopify-design/cart-add-an-agree-to-terms-and-conditions-checkbox/td...
There are also scriptable automation apps like usemechanic to build more robust custom scheduling processes. Examples: https://tasks.mechanic.dev/?q=schedule
Example form scripting https://tasks.mechanic.dev/tutorial-trigger-a-task-from-a-contact-form-using-webhooks
If you need this process setup for you then you can contact me directly.
Please provide context: store url, theme name, post url(s) and any further detail.
Contact Info in signature.
Good Hunting.
1. Unfortunately at this time there are no metafield definitions feature for the shop/store object exposed in the admin for merchants to simplify the initial setup process. And metaobjects are not exposed in flow either.
