A solution is available, I guess. I created a user profile here just to ensure you guys don't tear your hair out

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1. The earlier observation that it was because of having a locked SIM is very likely correct. I refuse to forego that protection (my bank account is linked to a SIM card therein and cloning is rampant in my country, for unprotected cards). So that simply wasn't an option for me.
2. I booted into safe mode and found that the issue did not come up. (Safe mode only loads the android OS without extras and your installations running.) All seems to run well. Thus, nothing with the basic telephony system was wrong or the culprit: GOOD NEWS!
3. Booting back into normal mode, the issue started again (as I expected).
4. A key effect of the problem is that any attempt to apply the airplane mode triggers that "emergency" message with a concurrent inability to actually get the airplane mode to take effect.
6. To solve this
A) Use the tray pin to eject the SIM tray while the phone is still running.
B) The rogue scripts that are interfering thus go dormant.
C) Now, apply the airplane mode. It will work now. I was simultaneously connected to my Bluetooth headset (and listening to music) when I did this, so I don't know if this is also exactly needed to effect this "cure" (though I don't think so).
D) With the airplane mode active, reinsert the SIM tray. Wait for a while (I left it for about 3 minutes).
E) Remove the airplane mode and see if the signals would be restored without the "emergency" error.
I tested, severally, putting on the SIM data mode, reapplying and removing the airplane mode. No issues.
I suspect that the sequence described above may need to be repeated whenever one reboots (which, since done —by me anyways— just about once weekly, is bearable).