I saw this thread start just as I was dealing with yet another crisis. . .I'm glad to see a nice discussion came of it, because I thought it was an interesting question.
For those who are not organists and familiar with playing on different actions, I'd point out that different kinds of actions do often have a subtle effect on your music making. It is true that the sustained part of pipe sound does not vary (too much, though it can) from one action to another, the expressive part of pipe sound is its initial attack, and somewhat, its release. This is especially true in the case of instruments voiced in pre-19th century style, but is true even of Romantic instruments.
A well regulated mechanical action organ which is voiced sympathetically is quite sensitive to an organist's touch. Because the organist makes a direct kind of contact with the pallets under the pipes, the faster a key is pressed, the faster the pallet opens and often the more chiff there is in the attack. It takes some skill and practice to master this very subtle effect, and organists who are not trained or do not practice on mechanical instruments have trouble mastering it. Some mechanical instruments are so sensitive that playing with the fingers held above the keys and then striking the notes as with piano touch causes pretty bad pipe speech. These same things are true to some extent with how the pallets CLOSE. I fast release produces and ugly "thwap." This aspect of mechanical action technique is more subtle than the attack, and I believe takes more time to master.
When it comes to other types of action, the organist is relinquishing control of the initial pipe speech and the release to a mechanical device, with the trade off that it is easier to play large instruments which take a lot of energy to play if they are mechanical. It is true that there are good and bad examples of all of these kinds of actions, but it's also true that they do have certain characteristics in general. A sensitive organist, especially one who is accustomed to mechanical action, will tend to have preferences based on how particular actions tend to attack and release the sound. My personal opinion is that some electro-pneumatic actions are not too bad in this regard. I grew up playing tubular pneumatic instruments that were abominable, though I do know of one in Brooklyn that plays exceptionally well--it's reliable, and its attacks and releases are nicer than on any electro-pneumatic instruments I've played. The worst (in my opinion) but also the simplist is the direct electric action because the pallets are placed directly beneath the pipe toes. The pallets slam down quickly and the air puffs into the pipe in a hard manner that makes voicing difficult. Sometimes this issue is dealt with by increasing the thickness of the toeboard, and I have played one or two instruments that don't sound too bad, but this option is expensive and takes away from the most appealing aspect of direct electric action--it's cheaper, simpler, and allows a lot of flexibility in how the pipes are arranged. However, I'd say it has so many drawbacks that I have not heard many that I like.
As to cone action, that is very rare here in the US, and I have never played one. I've always been very curious about how they affect the playing of the instruments, how they feel under the organist's fingers, and what their musical effect is like.
