Q: for some reason my iPhone 5S does not show up as a device

Have you restarted your computer?

If iTunes doesn't recognize your iPhone, iPad, or iPod - https://support.apple.com/HT204095

Device not listed immediately after upgrade - https://discussions.apple.com/thread/6573744 - try restarting

Check you are using the original or Apple replacement cord. Some third party cables only transfer power and no data signal.

LincDavis July 2014 post on resetting sync services when iTunes not responding - https://discussions.apple.com/message/26308435#26308435

LincDavis June 2015 post on other things to try - https://discussions.apple.com/message/28397802#28397802

New phone not recognized when given exactly the same name as old phone. - https://discussions.apple.com/message/26760759#26760759 - delete all backups from old phone.

Device not recognized because of cable issue (although user initially assured it was a good cable) - https://discussions.apple.com/message/28002758#28002758

https://discussions.apple.com/message/29154537#29154537 - deleted iTunes application then re-installed

July 2016 Lawrence_Finch post - https://discussions.apple.com/message/30402529#message30402529 - dirty connection?