feat(booking): stepped public booking flow with service/staff selection

The public booking page showed a capped flat preview of slots with no way
to pick a service, day, or staff member. Replaces it with a stepped flow:
pick service, optional staff preference, pick a day, pick a time.

Backend: availability accepts serviceId/staffId/date filters (default
response unchanged for the embed widget), a new public catalog endpoint
exposes bookable services and active staff, and slots carry staffName.
Slot generation and booking creation now honor tenant-wide and per-staff
blocked days; appointment requests must fit inside a matching
availability rule and reference tenant-owned, active staff and locations.

Generated with [Devin](https://devin.ai)

Co-Authored-By: Devin <158243242+devin-ai-integration[bot]@users.noreply.github.com>

packages/api-client/src/generated/types.ts

@@ -607,6 +607,22 @@ export interface paths {
         patch?: never;
         trace?: never;
     };
+    "/v1/public/tenants/{tenantSlug}/catalog": {
+        parameters: {
+            query?: never;
+            header?: never;
+            path?: never;
+            cookie?: never;
+        };
+        get: operations["getPublicCatalog"];
+        put?: never;
+        post?: never;
+        delete?: never;
+        options?: never;
+        head?: never;
+        patch?: never;
+        trace?: never;
+    };
     "/v1/public/widget.js": {
         parameters: {
             query?: never;
@@ -1033,6 +1049,11 @@ export interface components {
             tenantSlug: string;
             timezone: string;
         };
+        PublicCatalogResponse: {
+            services: components["schemas"]["PublicService"][];
+            staff: components["schemas"]["PublicStaff"][];
+            tenantSlug: string;
+        };
         PublicConfig: {
             /** Format: uri */
             apiUrl: string;
@@ -1040,6 +1061,18 @@ export interface components {
             environment: string;
             neonAuthEnabled: boolean;
         };
+        PublicService: {
+            durationMinutes: number;
+            /** Format: uuid */
+            id: string;
+            name: string;
+            priceCents: number;
+        };
+        PublicStaff: {
+            displayName: string;
+            /** Format: uuid */
+            id: string;
+        };
         Service: {
             /** @description Blocked time after the appointment (0-480) */
             bufferAfterMinutes: number;
@@ -1136,6 +1169,7 @@ export interface components {
             serviceId?: string | null;
             /** Format: uuid */
             staffId?: string | null;
+            staffName?: string | null;
             /** Format: date-time */
             startsAt: string;
         };
@@ -2674,7 +2708,14 @@ export interface operations {
     };
     getPublicAvailability: {
         parameters: {
-            query?: never;
+            query?: {
+                /** @description Return all slots for a single tenant-local day (YYYY-MM-DD) instead of the capped preview */
+                date?: string;
+                /** @description Restrict appointment slots to a single service */
+                serviceId?: string;
+                /** @description Restrict appointment slots to a single staff member */
+                staffId?: string;
+            };
             header?: never;
             path: {
                 tenantSlug: string;
@@ -2694,6 +2735,28 @@ export interface operations {
             };
         };
     };
+    getPublicCatalog: {
+        parameters: {
+            query?: never;
+            header?: never;
+            path: {
+                tenantSlug: string;
+            };
+            cookie?: never;
+        };
+        requestBody?: never;
+        responses: {
+            /** @description Bookable services and active staff for a tenant booking page */
+            200: {
+                headers: {
+                    [name: string]: unknown;
+                };
+                content: {
+                    "application/json": components["schemas"]["PublicCatalogResponse"];
+                };
+            };
+        };
+    };
     getWidgetScript: {
         parameters: {
             query?: never;