I think you are asking too much from AppleTV or any streaming box. You want live sports and to watch TV that comes out on a daily basis. You also then say you love HBO but would not if you had to pay extra for it.
Well, AppleTV, Roku, FireTV, etc. do not offer those capabilities in total. There may be some alternate methods to watching what you enjoy, but it certainly will not be a direct replacement.
My recommendation would be:
1) 1st investigate the streaming device options. The big ones are AppleTV, Roku, FireTV, Google Chromecast, and Google Nexus. There are other boxes, such as Android devices you can get on eBay or Amazon, as well as setting up an HTPC.
2) Pick the one that likely covers most of what you'd like to watch, then determine which apps/subscriptions you'd be willing to pay for (HBO, Showtime, Netflix, etc.).
3) Consider adding an antenna for free over the air programming. Your reception will be completely dependent on your location. I get all of the major networks.
4) If you are happy with the antenna idea, consider a Tablo or HDHomeRun device. These are TV tuners that plug into the antenna for input and your home router for output. They are then accessible on the streaming device. In effect, live TV from your antenna is available through a streaming box's app.
All together, the above will allow you to cancel cable and have a single streaming box solution. The investment is based on the hardware you select, but would be $500 max, $200 on the low end. So you'd recoup that investment in a few months of being cable-less.