I successfully created a Pushover account on the Pushover.Net website, and all indications were that the KM Pushover action was working (because any text I sent using the KM Pushover action used one of the free 10,000 monthly messages) but not a single message created with the KM action was delivered to my Mac client until about 24 hours afterwards. Now they arrive instantly (~1 second) when I create/send a message with the KM Pushover action. I don't think I did anything on my Mac to have "enabled" the messages. But it appears that all the messages I sent yesterday have arrived today.
A 1 second delay is excellent. I'm very happy now. Yesterday I was not happy; I thought I was doing something wrong, but now I think it was not caused on my end. Either Pushover.Net needs 24 hours to activate an account fully, or their servers were down.
Now the next question I have is whether KM can "trigger" on an incoming notification from the Pushover app on my Mac. (Preferably with the text of the notification as the %TriggerValue%.) I suspect the answer is no. In that case, are there any alternatives like examining a system log file to see what notifications have arrived. If the answer to that is no, I may have to use some OCR wizardry to simulate a notification trigger.