>-- a site not being able to tell the location of your dns server, which is true
>-- your dns server not affecting how sites adjudge your location, which is not
Neither of these statements are definitively true/false.
1. Sites can most certainly tell the IP (and thus the location) of your DNS server. There are many sites that demonstrate this, just search for "dns leak test". Whether sites actually use this is another question.
2. Sites can serve different IPs (servers) depending on the DNS server, or even the client (through the edns client subnet extension). Some CDNs use this strategy to route requests to the closest server. However, this fact is a red herring when it comes to assessing whether "just change your dns to a UK server" is a viable strategy for getting bbc iplayer to work, because its geoblock checks based on IP of the http request, not through DNS.
There's also the question of "smart dns proxies"[1], which make it seem like all you're doing is "change your dns server to a UK server", but there's far more that goes under the hood than just changing your DNS server, because it's actually proxying your traffic as well. Changing your dns server to a uk server that isn't a smart dns proxy wouldn't get you pass bbc's geoblocks.
-- a site not being able to tell the location of your dns server, which is true
-- your dns server not affecting how sites can adjudge your location, which is not