Just got off the chat with a super helpful Apple Support person.
What's happening is the screen stopped working. Not sure yet if it's a hardware or software issue (probably hardware). The weird and loud sound it's blaring when you press and hold Vol Up + Side Button is the iPhone placing an SOS phone call and messaging your emergency contacts.
What to do trying to force a restart on your phone, and that's done by a new key combination. It's not Vol Up + Side Button. It's "Press and quickly release the Volume Up button. Press and quickly release the Volume Down button. Then, press and hold the Side button until you see the Apple logo." If you, like me, don't see the Apple logo after 10–60 seconds, then your screen has a problem.