Yes, you'd download some things you don't need to. Last I heard, they're working on an extension so the browser can tell the server what media it has when it requests the initial page so they won't get sent at all.
Didn't we all have this conversation years ago already with nocache query parameters etc.? Reminds me of number 6 in RFC 1925: "It's easier to move a problem around [to another protocol layer] than to solve it."
I still don't see how HTTP/2 Push is solving any real-world problems [1] besides Google being able to push ads so that it counts as an impression even if the client blocks ads.
[1] I don't count the slightly faster initial page load since I'm getting that junk on every page load via push even if I already have it, so I'm paying with my data allowance. If you want your site to load fast, just fix your goddamn bloat.