I don't think 1 additional line of code is much of a burden. If you put it in the URL you're forcing the server to ignore the Accept header that the browser sends and pay attention to some non-standard url substring instead.
Now I navigate off the page to a different site, and press the back button. Instead of the HTML page, I get the cached JSON response. Now if I change the ajax call to "example.com/objects/1.json" instead, that keeps the URLs separate and the browser won't cache the wrong response. Is there a better way to solve this?
[1]https://developer.mozilla.org/en/HTTP/Content_negotiation
Think of all the different "Accept" headers that clients can send you, each one of these will get an entry in your cache. And now add other headers that should also be in "Vary" like "Cookies", "Accept-Language", etc. and your cache will be virtually useless.
At least in python, making requests becomes quite a few more lines of code if you want to add headers. There's usually a short-hand command if you just want to get full data from an url and a "create-an-object, then set these parameters, then this, then open a connection then read what's there".
Mandatory headers are just a way to make life of a regular developer painful.
Accept: text/html,application/xhtml+xml,application/xml;q=0.9,/;q=0.8
This says "we would prefer one of these formats, but we'll take whatever you got".