"To apply the changes you must restart your computer"
It is just sad.
Both PulseAudio and PipeWire work best when running the underlying kernel drivers in "exclusive" mode, which means only one sound server can be running at a time. In order to ensure that works correctly, you need to:
1. Close all applications
2. Shut down PulseAudio and prevent it from auto-restarting
3. Start up PipeWire, and then restart your desktop environment if you want any hope of the volume applet working (or figure out which of the processes to kill if your volume applet in another process)
4. Start up your applications again.
5. There might be some applications that have tried to play audio that you were unaware of! Like your file manager, which can play a custom "bell" for mandatory audio alerts when trying to type in a place that doesn't accept characters. Oops! Better close that down too and start 1-4 over again.
Restarting the computer is an easy and safe way to apply these changes. That people didn't follow all these steps in this exact way meant the computer started to break in strange and unobvious ways, not that it was ever supported.
If you want to help out designing smoother ways to change out core system components without reboots, feel free to do that. The people involved might not find it worth the trouble to fix, though. I think it's just sensible to restart the system.
https://wiki.archlinux.org/title/PipeWire
Your session environment might also need to be aware of how to interact with PipeWire, which by default is via pulsaudio compatibility interfaces.
Required (practically, probably pulled in with just pipewire):
pipewire pipewire-media-session pipewire-pulse
Recommended:
pipewire-alsa pipewire-jack lib32-pipewire lib32-pipewire-jack
You might also need to reconfigure your audio settings, but that is unfortunately desktop session specific. (E.G. KDE has it in one place, gnome desktops another, etc)