Wireless Internet, Really.

I would like to know why wired internet still exists. Currently I need to pay for internet for my home, it is reasonably cheap but only works within the confines of my home. When I leave my phone I often use the same devices, my phone or iPad for example, on mobile internet. Its much more expensive than my home internet but I can use it anywhere. My question is why do we differentiate between the two? Clearly the future is wireless internet. Home phones are a dying breed because people can just call my mobile all the time and I can use it where ever I am even in the house. Its a but more expensive for me than a landline but it makes more sense. I believe the future is in mobile internet. Providers would have to build up their infrastructure for this but not so much and they would never have to worry about running cables into homes anymore. Why can't Verizon just install a small tower in my neighborhood to replace the internet switch box already there? Each neighborhood could be supported on a small tower and larger towers would only be needed in rural areas. They would sell me a modem like they do now for my home to connect my devices to. I would pay them about the same as a mobile data plan and regular internet for like 250GB of LTE. I can use it wherever. End of story.

So back to my original question, why does wired internet still exist?

P.S. That isn't just a hypothetical question, I would like to know of my idea is feasible from a technology and business standpoint.