iPhone keeps shutting down

Give these steps a try

If your iOS or iPadOS device unexpectedly restarts or displays the Apple logo or a spinning gear icon – Apple Support (UK)

That’s a symptom of a failing battery. Make a genius bar appointment to get the battery replaced, or better yet, get a newer phone that will continue working as a phone going forward. All of the US networks are shutting down the networks that the 5S can access; AT&T already has, and the other carriers will be doing so later this year.