No, virtual memory and client/server computing do not drive multi-tasking. Virtual memory helps make it work better, but with sufficent real memory, you do not need virtual memory.
Multi-tasking is needed because people want to do several things at once, or want to switch quickly between them. Windows at least since 2.11 has supported some level of multi-tasking. It was cooperative, but it was multi-tasking.
NT was the first Windows version that had "real" multi-tasking.
A classic and easy to understand example of why humans want multi-tasking is that they want to play a music file, have their email program downloading email, all while they surf the web using FireFox.