It's not like it hasn't already been done -- sort of.
I wish I could recall exactly which mission it was, but I recall quite well an occurrence where the Falcon 9 engines started up, then shut down just prior to the hold-downs coming off. After a short analysis and quick fix, the count was recycled and the launch proceeded that night (IIRC), late but still within the launch window.
I believe the F9 sequence you are describing was prior to subcooled LOX...
Yes, I'm positive it was. I'm also pretty sure it was a Falcon 9 v1.1, prior to the current FT version that, we are told, should just be called Falcon 9...
Anyone remember which mission it was? I'm pretty certain it wasn't a CRS flight, they have very short launch windows. Maybe Cassiopeia? It was definitely pre-stage-recovery days.
SES-8, though based on a perusal of the "SpaceX launch log" thread, I think your memory may be playing tricks on you a bit. SES-8 got through start-up, but didn't release. They then recycled and attempted a second countdown. But that second countdown was scrubbed due to contamination of the ground side TEA-TEB. It then launched successfully a few days later. So, the gist of your point was still valid. They got through engine start-up, held the launch, and then recycled to try again within the same window. But it didn't launch on the second attempt either. That's the only launch that I could find that might fit your memory. It's also possible that the log didn't catch this somewhat unusual type of attempt.
Somewhat interestingly, SES-9 did pretty much the same thing. On the launch attempt with the wayward boat. They recycled after the range fouling and got through start-up but held for low thrust. They then considered another attempt in that same window, but couldn't because of the subcooled LOX warming too much. But if they hadn't had to worry about the warmer LOX, I think they would have recycled and tried again. I remember them taking a fair bit of time after the low thrust hold to finally scrub the attempt.