I do not have home phone or cable and I use the internet. as far as most programs/news/shows being on Netflix or Amazon Prime, it depends on what you watch.
You did not mention your location but where I live, local news is streamed so I can catch the weather but if you want to watch the latest episode of your favorite show, forget it, they will not be immediately streamed. In some cases you can go to the channels website and stream them a few days later or you will have to wait till the season is over and then hope it will be picked up by a streaming service. All of the above is based on using a computer as a streaming device.
if you have the cheapest internet package, you could have problems streaming depending on the speed. Some streamers have to bump up their package to keep their shows from buffering.
you will have to figure out how to get the internet to all of your tvs. I use cheap walmart computers in order to get the local streaming news and other shows not available on streaming boxes. they paid for themselves in six months.
if you dont need the local news and weather, then roku, firetv, chromecast and one of the other streaming boxes would be all you need would do. netflix and amazon prime are good services, in addition, you can subscribed to different video services like shutter, full moon, contv, showtime and others.
finally if you are into sports, there could be a disadvantage here. You can subscribe to some sport packages through the streaming box, it will be limited and may not get local favorites.