For those of you who don't already know this, I'm about to make it abundantly clear that I know not of what I speak...
Now that's out of the way, I can see how turf health and green speed would be forever bouncing back and forth. Healthy turf would I think enable the greenkeeper to cut the grass shorter and make the greens faster. the shorter grass then making the turf less healthy, so the grass has to be allowed to grow more, making the greens slower, but the turf healthier. Rinse and repeat.
On another note, when I was growing up, my club had very firm greens at times. Pitching wedge landing front third from the fairway wouldn't stop on the green, even with a balata ball. They rolled at maybe 8 on the stimpmeter on a good day. More likely 7-7.5. Conversely I have played other places where you can back up a 3 wood, but they run well above 10. I don't think firmness has much to do with green speed. Look at Open Championship venues, where sometimes it's bone hard (Muirfield a couple of years ago), but the greens aren't that quick - they can't be lest the wind blows.