OTA may have failed, leaving phone with "No Service," activation fails

iPhone 5s/32gb which was on the latest IOS version

1. Phone started showing “No Service” where the operator logo used to be. Allowed an hour or so to go by as I was in no hurry to get calls. No change after around an hour. Restarted the phone. No service.
2. It was late in the evening, so I left the phone off all night, powered on in the morning, got a signal, made an outgoing call, received a call. A few minutes later, “No Service.”
3. Rebooted the phone, got a system dialog “Cellular Modem Update Failure” or words to that effect. The support page suggested power cycling etc and if all else fails, to connect to iTunes and fully restore the phone.
4. Did all that. During this process, the “No Service” text gained a triangle with an exclamation mark. After this point, iTunes refused to proceed since it could not get Activation Data from the phone.

I have reason to believe a cellular modem update failed the previous night, leaving the modem firmware in an inconsistent state.

For whatever reason, iTunes does not seem to restore modem firmware?

Took the phone to Apple service. Without even looking at it, they asked for 2/3 the price of an iPhone 6s 32gb to replace the 5s.

Any pointers? Is it possible to get this phone back up to the point were I can use it at least as an iPod?