Screen mirroring is part of airplay, and every iPad has it, being is a wireless protocol used to stream media and mirror the display.
The first part of my response was merely pointing out the differences between the TV app and the Apple TV hardware, as you were commenting on the fact that the Apple TV menu was not showing up on your iPad and I was trying to point out why that is not possible.
The Apple TV menu is showing up on your TV. You just happen to only see two icons because it is not properly connected to the network. The device is very much plug and play, however it is still dependent on a proper network connection and when that is not received proper troubleshooting needs to occur. I have had every model and it works exactly in that matter, as does it for the vast majority of users. However, just as with any device those steps need to occur to investigate a problematic connection. Connecting to multiple TV's will not change that. I advised to use a different ethernet cable to rule out a possibly faulty one, checking the router as that port is required in order for functionality, and to check the status using a diagnostic report to verify the level your network is currently operating at.
This is a user support forum and the purpose is to help users troubleshoot when issues occur. If you are not sure of something then you can ask more questions, however if you're not intent on doing any further investigation then i'm not sure of the purpose for you posting here. To reiterate, this would apply to any network device when issues occur so perhaps you should rethink buying any such device if that is your stance. Also, the 3rd gen model that you have was discontinued in 2016, and long past the period for return.