My iPhone cannot connect to the internet on cellular data and wifi

See my response to this question on the Apple discussions.

https://discussions.apple.com/thread/5770041?tstart=0

Basically, use a static configuration for your WiFi Network, but leave the Router IP address blank. The LTE icon will be displayed, and the mobile network used for all non-local traffic (ie, Internet) but you will still be connected to the WiFi network can can use all local LAN resources.

Edit: I also notice you said use it as your primary Internet connection. I originally assumed that you meant only for the iPhone. If so, then my response above should work for you. If you meant use it as your primary Internet connection for all other devices (laptops, etc) in your home, then of course the only way to do that IS to use your iPhone as a hot spot. That's kind of the definition of a hotspot.

This is possible, as shown here: https://discussions.apple.com/thread/6448960?start=0&tstart=0

As the link describes, use a static IP in the wifi network but only fill in the IP and Subnet Mask fields.

Edit: this is also answered here: https://apple.stackexchange.com/a/174722/120252