Deprecate currentBridge API and drop its last internal uses (#59428)

Summary:

`+[RCTBridge currentBridge]` always returns nil and `setCurrentBridge:` is a no-op, so remove the two `RCTInstance` call sites and their now-dead private-header imports. Keep the declarations for Expo's Swift `RCTBridge.current()` caller and mark them deprecated.

Changelog: [Internal]

Differential Revision: D127265579

Author: christophpurrer

packages/react-native/React/Base/RCTBridge+Private.h

@@ -16,8 +16,9 @@ RCT_EXTERN void RCTRegisterModule(Class);
 
 @interface RCTBridge ()
 
-+ (instancetype)currentBridge;
-+ (void)setCurrentBridge:(RCTBridge *)bridge;
++ (instancetype)currentBridge __deprecated_msg("This API will be removed along with the legacy architecture.");
++ (void)setCurrentBridge:(RCTBridge *)bridge
+    __deprecated_msg("This API will be removed along with the legacy architecture.");
 
 /**
  * This method is used to invoke a callback that was registered in the

packages/react-native/ReactCommon/react/runtime/platform/ios/ReactCommon/RCTInstance.mm

@@ -13,7 +13,6 @@
 #import <React/NSDataBigString.h>
 #import <React/RCTAssert.h>
 #import <React/RCTBridge+Inspector.h>
-#import <React/RCTBridge+Private.h>
 #import <React/RCTBridge.h>
 #import <React/RCTBridgeModule.h>
 #import <React/RCTBridgeModuleDecorator.h>
@@ -329,7 +328,6 @@ - (void)_start
   bridgeProxy.jsCallInvoker = jsCallInvoker;
   bridgeProxy.performanceLogger = _performanceLogger;
   _bridgeProxy = bridgeProxy;
-  [RCTBridge setCurrentBridge:(RCTBridge *)bridgeProxy];
 
   // Set up TurboModules
   _turboModuleManager = [[RCTTurboModuleManager alloc] initWithBridgeProxy:bridgeProxy