Couple comments:
1) It doesn't surprise me that it primes the pump for a few seconds upon turning on the car, but I sure as heck can't hear ANYTHING whatsoever. In fact, I kind of assumed that was the behavior. So when I've been trying to "prime" the system, I turn on the ignition, wait a few seconds, turn it off, turn it back on wait a few seconds, turn it off, etc, maybe 4 times before trying to start. I haven't had a hard start while doing this, but I haven't tried that many times that I'm convinced this is correcting the issue.
2) I'm starting to suspect its not the pump:
On two different occasions, I drove the car, shut the car off for 5-10 minutes and tried to restart it, and it took quite a bit of cranking to start. The car also shuddered quite a bit when cranking, in a very unnatural way. On one occasion, after parking for about 35 minutes, I went to restart it, and it took a bit of cranking. I stopped cranking, and then a couple seconds later tried again, and the car started immediately. I'm going to try to see how repeatable it is.
Still no error codes, I'd think if the fuel pump was weak, I'd be getting some sort of code, like inadequate fuel pressure, or too much long term fuel trim, etc. I think a weak check-valve would be more likely, but if it depressurizes the lines in 5-10 minutes on a "hot" car, then it should ALWAYS be failing when I'm cold starting it, like in the morning or when leaving work.