How to send event from MainActivity to React Native in new architecture (bridgeless mode)?

I have following implementation (I've skipped imports and some parts):

class MainActivity : ReactActivity(), DefaultHardwareBackBtnHandler, PermissionAwareActivity {
    var communicationModule: CommunicationModule? = null
    

    fun sendMessageToNative(message: String) {
        val manager = (application as MyApplication).reactNativeHost.reactInstanceManager
        val reactContext = manager.currentReactContext

        if (reactContext != null) {
            val module = reactContext?.getNativeModule(CommunicationModule::class.java)
            if (module != null) {
                module.emitOnPush(message)
            } 
        } else {
            manager.addReactInstanceEventListener(object :
                ReactInstanceManager.ReactInstanceEventListener {
                    override fun onReactContextInitialized(context: ReactContext) {
                        val module = context.getNativeModule(CommunicationNewModule::class.java)
                        if (module != null) {
                            module.emitOnPush(message)
                        } 
                    }
            })
            manager.createReactContextInBackground()
        }
    }

    override fun onCreate(savedInstanceState: Bundle?) {
        super.onCreate(null)
        // ...
    }

    override fun onBackPressed() {
        super.onBackPressed()
        sendMessageToNative("MY_MESSAGE") // this doesn't work (no event on RN side)
    }

    override fun onNewIntent(intent: Intent) {
        super.onNewIntent(intent)
        // assume this is not empty
        val myMessage = intent?.extras["message"]
        sendMessageToNative("MY_MESSAGE") // this doesn't work (no event on RN side)
    }
}

and Turbo Module (NativeCommunication.ts)

import type { TurboModule } from 'react-native';
import { TurboModuleRegistry } from 'react-native';
import type { EventEmitter } from 'react-native/Libraries/Types/CodegenTypes';

export interface Spec extends TurboModule {
    onPush: EventEmitter<string>;
}

export default TurboModuleRegistry.getEnforcing<Spec>(
    'NativeCommunication',
);

and Kotlin's (CommunicationModule.kt)

@ReactModule(name = CommunicationModule.NAME)
class CommunicationModule(private val reactContext: ReactApplicationContext) : NativeCommunicationSpec(reactContext) {

    override fun getName() = NAME

    companion object {
        const val NAME = "NativeCommunication"
    }
}

and finally setting listeners on React Native side (e.g. in some page or MyPage.tsx)

const MyPage = ({ navigation }: MyPageProps): JSX.Element => {
    const listenerSubscription = useRef<null | EventSubscription>(null);
    const callbackMethod = (message: string) => { console.log(message) }

    useEffect(() => {
        listenerSubscription.current = NativeCommunication?.onPush(callbackMethod);
        return () => {
            listenerSubscription.current?.remove();
            listenerSubscription.current = null;
        }
    }, []);
}

PROBLEM: events trigered in onNewIntent and onBackPressed do not arrive to React Native.

My implementation works (events arrive to React Native) only when triggered from

fun MainActivity.newSessionTimer(): CountDownTimer {
    val sessionTimerFinishAfter = inMilliseconds(30, TimeUnit.Second)
    val sessionTimerTickEvery = inMilliseconds(30, TimeUnit.Second)
    return object : CountDownTimer(sessionTimerFinishAfter, sessionTimerTickEvery) {
        override fun onTick(millisUntilFinished: Long) { /** no-op */ }
        override fun onFinish() { sendMessageToNative("MY_MESSAGE") }
    }
}

GOAL: Be able to send messages from onNewIntent and onBackPressed and receive them on MyPage.tsx