That's not correct - while any HTTP/2 capable servers along the path
may implement server push, the most important one is the closest server to the user (i.e. the CDN). The only requirement is that the intermediate servers must not strip out the server push headers/tags, so that the edge server knows what to actually push.
It's quite possible to have your own server provide HTTP/1.1 and still take advantage of server push, and this is a usecase supported by Akamai[1], Cloudflare[2], and I suspect other CDNs (can't find details) - you may get better performance using HTTP/2 on that intermediate link, but that's less likely to be because of server push.
[1]: https://blogs.akamai.com/2016/04/are-you-ready-for-http2-ser...
[2]: https://blog.cloudflare.com/announcing-support-for-http-2-se...