From the screenshot you've provided, it seems like your replication agent is disabled. I can see this because of the "red" icon. Typically when a specific replication agent is enabled, the icon should become "green".
Okay, next step. Assuming that you've re-configured the replication agent, and it still not show up as enabled. You need to check these things (please restart your AEM instance before proceeding to optimise your best testing environment):
Error.log, tail the error.log in .../crx-quickstart/logs/error.log. What kind of errors do you see while trying to enable the specific replication agent?
Run Modes; are you configuring the correct replication agent within the current context of the environment? To simplify things, use /etc/replication/agents.author/* on "author" run mode and /etc/replication/agents.publish/* on "publish" run mode. Let's not complicate things by adding custom run modes.
Once your agent is running, you should be able to see logs from the replicating agent coming in. Even if your dispatcher environment is not set up, when your replication agent is trying to perform a task, and if it fails of successes, logs should be shown in the "view logs" button within the configuration page of the replication agent.
If you really want to test this mechanism from out of the box, green-field environment, in your local desktop, perform tasks below (by default, the replication configuration on the author instance will publish to localhost:4503):
Startup a new AEM "author" instance, you can simply double click on the quickstart JAR (let's keep it with default settings, with samples,localhost:4502).
At the same time, start up a new AEM "publish" instance, you can simply double click on the quickstart JAR (let's keep it with default settings, with samples, localhost:4503).
Once both your local AEM author and AEM publish instances are running, you can simply replicate anything, and you should see a successful replication. (because by default, the replication to localhost:4503 publish agent is enabled on the author)
Try clearing all browser cache & cookies, check Disable Cache checkbox in Network tab of Developer Tool, close all browser windows and open a new one in Incognito Window. Probably, try another browser, such as Mozilla Firefox in Incognito Mode to verify it again.
Also, try to recompile JSPs & rebuild clientlibs, if it still doesn't work.