The issue of leaving a class D on all the time vs. turning it off is probably more dependent on what the Class D amp is.
I have had experience with Class D amps which use ICE output modules and I never liked them. Those amps definitely seemed to smooth out if left on to warm up for an extended amount of time vs. just turning the system on and using it. However I found the upper frequency range to be harsh even after warmed up; it was better, but it grated on me with use.
I am currently using a pair of Anthem M1 mono block Class D amps. This is a whole new world vs. the ICE based amps. The M1's do not use any devices such as the ICE modules, it's all proprietary design and liquid cooled. The M1's play super smooth at all times, I don't need to warm them up and I don't leave them on; it makes no difference that I can hear. I prefer to leave them off and unplugged from the 240volt outlets when not in use just in case of storms or surges. I do not trust surge protection devices as I have had major destruction due to lightning hits.
I also own a Parasound A 51 which runs in Class A up to a certain output power before is swaps over to Ab. That amp seems to like some warming up to smooth out, but even when at a fully warmed up state of operation, it doesn't come close the M1's in performance. I use it now to drive surround/rear/middle channels when using streaming music or HT films. For two channel stereo, the M1's do the heavy lifting.