It's not useless -- you have a cell phone issue of some kind. Just checked with wife who says she is always on our ATT cell phone network when out shopping and accesses the Shopping List. Works fine. You need to turn off your home Wi-Fi network on phone when at home to test this so you are not on it and be on your cell network only. You should be able to access your Alexa app and see Shopping list as well as other app features.
This is what I just did at home, and you should be able to do same:
Turned off iPhone Wi-Fi, so was on cell LTE network only.
Opened Alexa app and accessed Home, then Shopping List and added beer to the list by tapping in the "add it" box, told app (via phone mic) by saying "beer" and tapped plus sign in box. and beer was added to the list!
You can also do this if connected to store's Wi-Fi or a friend's. or hotspot's Wi-Fi. The only requirement is that the Alexa app needs to access the internet, and it will under all these conditions if they are met.
If you can't do this, you have an issue, so contact Echo Support at 1-877-375-9365
.