I figured it out. ifplugd clears all addresses when bringing up an
interface:

void interface_up(int fd, char *iface) {
...
        ((struct sockaddr_in *)(&ifr.ifr_addr))->sin_addr.s_addr =
INADDR_ANY;
        if (ioctl(fd, SIOCSIFADDR, &ifr) < 0) {
...
}

Since wg-quick sets the address before bringing it up, this running in
between is problematic. One workaround in wg-quick would be for me to
reverse the order of setting the IP and bringing it up, but this introduces
other races I'm not very fond of introducing. So, rather, we should address
the larger question: why on earth is ifplugd being started when wg0 is
added?

Sep 10 23:57:51 janus ifplugd(wg0)[14109]: ifplugd 0.28 initializing.
Sep 10 23:57:51 janus ifplugd(wg0)[14109]: Using interface
wg0/00:00:00:00:00:00
Sep 10 23:57:51 janus ifplugd(wg0)[14109]: Using detection mode: IFF_RUNNING

What causes it to be launched here? Digging a bit deeper, it looks like
ifplugd is being launched by a udev rule which calls a Debian file called
ifplugd.agent. It is in here that unholy hotplugging occurs.

I don't actually have a Debian system running to fish around and see, but
my guess is that you have a file /etc/default/ifplugd that has in it
HOTPLUG_INTERFACES=all. If you change this to HOTPLUG_INTERFACES="wlan0
eth0" or something more restrictive, things might work better. Just a guess.