My oldest laptop at home is this one:
That's a 4GB RAM, Core2Duo 2009 laptop with Intel HD graphics and is running Windows 10 since it was offered and I need as many W10 machines as possible for testing. By way it's been great!
It streams Amazon Prime, Hulu and more to a 1080p HDTV just grand. I can't guess why the machines you have tried have issues but a Vista machine we had (past tense) was a slug. It did OK on Linux but that's technical so moving on.
Best Buy may not be the place to try this as the network is swamped with all the things there. Here I get 2 video streams all the time on a basic Comcast internet plan. A renter does Netflix and I'm watching something on our Amazon Fire TV.
But again, I did upgrade our router to a newer dual core router so I have that going for me.
That out of the way, I don't think this is so much a laptop issue as the 2009 laptop does streaming just fine but network connections or VIsta.
Let's look at http://www.amazon.com/Best-Sellers-Electronics-Laptop-Computers/zgbs/electronics/565108/ today.
The first 1080 laptop at $799 is http://www.amazon.com/Dell-Inspiron-i7559-763BLK-Full-HD-GeForce/dp/B015PYYDMQ/
There's another Dell down the page at $512 but I think you want 1080.
I am looking to buy a laptop with the primary goal of watching live sports. My current computer, an old Dell desktop that uses Windows Vista, Intel processor about 2.5, actually streams baseball fairly well; hockey not so much. It doesn't hang up much, and the picture, for baseball at least, is fairly sharp.. My Ipad Air 2 streams baseball games that are very sharp and clear, almost never hangs up. I can't complain, except for the screen size. But when I go to Best Buy or try other people's computers to stream live sports, I find the picture blurry, even for baseball where there is not much continuous movement. I don't know why.
So, I am open to recommendations. My budget ceiling is about $1800 or so. I want the most crystal clear picture in a laptop for streaming live sports in that price range. Of course, I will also use it for YouTube, reading online newspapers, some word processing etc. I am not a gamer. I am also not a computer whiz in any respect.