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.
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.