Good day Telbuoy,
Thank you for contributing to the Apple Support Communities.
It sounds like your iPhone 5s is not connecting to your home Wi-Fi network. You've done a good job of isolating this behavior to your iPhone and home Wi-Fi router.
There are a few more steps that I'd suggest trying next:
Make sure your iPhone is updated to the latest iOS version using the steps in Update the iOS software on your iPhone, iPad, and iPod touch.
Make sure your router is using the settings in Recommended settings for Wi-Fi routers and access points.
Update your Wi-Fi router with the latest firmware and make sure the router supports your Apple product. For more information, contact the manufacturer of your router. (from If your iPhone, iPad, or iPod touch won’t connect to a Wi-Fi network)
Best Regards,
Jeremy