Followup 2: On a hunch, I copied the Client folder from the Version Cue CS4 folder into the Version Cue CS3 folder and changed the name from 4.0.0 to 3.0.0. Surprise, it works. Apparently there aren't enough differences in the contents of the folder to bother Illustrator CS3.
Still, the question remains, why didn't the CS3 Web Premium installer include a Version Cue Client folder? And why, way back in 2007 when I first installed and used Illustrator CS3 did the same problem not arise? I should mention it again that CS3 Web Premium would not install off of the original DVD this time around. It had all sorts of problems, including a demand for the CS4 Web Premium disc. I had to go online and get the download version, which worked without the same problems—other than the ones I'm discussing here.
There actually is a Version Cue CS3 Client dmg file in the Payloads folder. I expanded it and copied the included Client folder to the Adobe folder in Applications support, replacing the CS4 Client copy. But it is incomplete and Illustrator generated the same missing component error message. So I had to restore the CS4 Client copy.
Other apps in the CS3 suite work without a problem, including Dreamweaver, Flash and Fireworks.
Followup 3: I worked on my friend's Mac today, following the steps, with a few variations, that worked for me. She's using an old 17" Macbook Pro running OS X 10.6.8 Snow Leopard.
To begin with I tried to uninstall Adobe CS3 Design Premium. She did not have an Adobe Installers folder from which I could launch the Add or Remove Adobe Creative Suite CS3... app. So I tried to use her CS3 installer to do it. As on my system, it failed.
So I used the CS3Clean Script (Adobe - Adobe CS3Clean Script) to do the uninstall. Be warned, this script is extremely thorough. Follow the instructions carefully if you need to use it. It ran OK on her system, Snow Leopard. It will not run on mine, OS X 10.9.5 Mavericks.
Then I prepared to run the Creative Cloud Cleaner Tool (Use the Adobe Creative Cloud Cleaner Tool to solve installation problems), restarting the computer after uninstalling CS3. This, too, should be used carefully. The only item I needed to remove was the hosts file. After this I installed CS3, not from her CS3 DVD, which probed problematic, as mine did, but with the Adobe CS3 Design Premium installer Adobe makes available, along with other CS3 Studio versions, at Advanced solutions to connection errors with Adobe Creative Cloud and Creative Suite applications.
After the installation finished, I launched Illustrator CS3. The same "expired" message appeared, but this time when I tried to reauthorize the suite, the web connection worked and the authorization succeeded. As on my system a message appeared saying that registration failed do to the lack of a connection, but this registration is not critical to using CS3 so I clicked Never Register, the dialog disappeared and Illustrator continued to launch as expected. There was no fatal error, as I experienced, so apparently the Design Premium installer included the Version Cue Client files that Web Premium did not (see Followup 2 above).
I cannot speak for anyone else posting here, but this procedure has worked for me now on two different systems. It took me nearly a week to gather the necessary information, or rather, for that information to be put forward by Jeff Wright. Thanks, Jeff. The process was arduous, with much confusion and many misdirects, but, as they say, all's well that ends well. I hope others having similar problems will find this summary useful. Be sure and let us know if you do—or if you do not.