How does it recognize the device I'm using?
There are several ways for websites to determine which browser and kind of device you are using. Modifying your "User Agent" profile won't necessarily bypass them. See this related question from security.stackexchange. You can use the "What Is My User Agent" website to determine what you are sending out.
With Safari, try killing the process Safari Web Content (from Activity Monitor). Also, try clearing your cookie cache - including "supercookies".
The cbsnews live stream video plays on my iPhone 4S running iOS 9.3.5 so after clearing your cache and killing Safari Web Server, you could try a User Agent profile like:
Mozilla/5.0 (iPhone; CPU iPhone OS 9_3_5 like Mac OS X) AppleWebKit/601.1.46 (KHTML, like Gecko) Version/9.0 Mobile/13G36 Safari/601.1
...this User Agent profile was obtained from my iPhone using the "what's my user agent" website.
Also from this How-To site you could try these or similar "User Agent" profiles:
What is a User Agent Anyway?
When Safari visits a website, it will send a string of text such as this:
Mozilla/5.0 (Macintosh; Intel Mac OS X 10_10_2) AppleWebKit/600.3.18 (KHTML, like Gecko) Version/8.0.3 Safari/600.3.18
This tells the web server that this particular user is running Safari 8 on a Mac running OS X 10.10.2.
It will obviously be different per the operating system and web browser. > A computer running Windows 7 and Internet Explorer 10 would appear as such:
Mozilla/5.0 (compatible; MSIE 10.0; Windows NT 6.1; Trident/6.0)
but I think it is best to test a known, working user agent profile on your system.