Is there a way to test whether an iPhone (5s) is failing to work on certain frequencies?
The Genius bar may be able to help you by running diagnostics, but you can also try that yourself with this project that MITMs the diagnostics connection and dumps all data into an archive, you can try poking around in in and see if you can find anything relevant in it.

Another solution would require control of the mobile network but then you can definitely see why the phone isn't connecting to the desired networks (if the SIM allows it and the phone still doesn't connect then it's a phone failure).