I believe the polling implementation was brought into the main exec file, and still exists here:

I don't know whether that means we should be calling it from Java or not... clearly this has been broken for a long while

I believe the polling implementation was brought into the main exec file, and still exists here:

cordova-android/cordova-js-src/exec.js

Lines 120 to 132 in 5a2c50d

	function pollOnce (opt_fromOnlineEvent) {
	if (bridgeSecret < 0) {
	// This can happen when the NativeToJsMessageQueue resets the online state on page transitions.
	// We know there's nothing to retrieve, so no need to poll.
	return;
	}
	var msgs = nativeApiProvider.get().retrieveJsMessages(bridgeSecret, !!opt_fromOnlineEvent);
	if (msgs) {
	messagesFromNative.push(msgs);
	// Process sync since we know we're already top-of-stack.
	processMessages();
	}
	}

I don't know whether that means we should be calling it from Java or not... clearly this has been broken for a long while

Might be able to expose that method by added it to androidExec:

androidExec.pollOnce = pollOnce;

Then instead of deleting the line, update it to:

            if (!willSendAllMessages) {
                sb.append("window.setTimeout(function(){cordova.require('cordova/exec').pollOnce();},0);");
            }

But I don't know what the behavior it will have after adding a pollOnce call to something that never ran for 12 years...