Piano strings are fairly linear devices. Initial transient effects in linear systems tend to die out pretty quickly. As a pretty neat demo of this, here are some waveform images of some of the keys in the temperament of the same piano I captured with Reaper, which I described in my discussion of tuner training software. Note that these are recordings of true single piano strings, not simulated. The name of each track, at the very left of the track just above the volume adjustment sliders, are the names of the notes represented in each track.
The top of the image shows the time axis in seconds, so the whole graph is only about 2.4 seconds long. You can see that the transient effects vary from string to string, which I don't understand. May be poor rebound from the string on those notes, as you mentioned. But those effects, whatever they are, are only less than a tenth of a second. The worst shown here is on F4, the 6th trace from the top. You can also see from the traces that the higher strings decay at a much faster rate than the lower strings. Pretty dramatically so.
And just for ya-has, heres some of the waveforms zoomed way in, so that you can see the actual time representation of what the microphone picked up.
You can see from these that some of the keys have a lot more higher partial content than the others. Possibly because I've never voiced the hammers since I installed them. Some are significantly quieter than the other as well. I noticed this happening when I was recording them, although I couldn't detect much difference in volume through my ears. Don't know what to say about that.
The bottom set of notes are not all the same as the top set but most of them are. Of the ones that are common to both sets, you can see that the ones with the strongest transient effects in the top also appear to be the least well-behaved ones when you zoom way into them in the bottom set. Pretty cool, huh? I wonder if the ones with the stronger transient effects could be because of poor string terminations, or perhaps loose hammer flanges, or who knows what?
I'd say that there is probably a lot that could be learned by studying these things. I just started looking at them in the past week or so. It will be interesting finding where this brings me, if I stick with it.